Fractal Analysis - Basic Concepts and Applications [1 ed.] 9789811239434, 9789811239441, 9789811239458

Citation preview

FRACTAL ANALYSIS Basic Concepts and Applications

12345_9789811239434_TP.indd 1

24/1/22 9:57 AM

B1948

Governing Asia

This page intentionally left blank

B1948_1-Aoki.indd 6

9/22/2014 4:24:57 PM

Series on Advances in Mathematics for Applied Sciences – Vol. 91

FRACTAL ANALYSIS Basic Concepts and Applications Carlo Cattani

University of Tuscia, Italy

Anouar Ben Mabrouk

University of Monastir, Tunisia & University of Kairouan, Tunisia University of Tabuk, Saudi Arabia

Sabrine Arfaoui

University of Monastir, Tunisia University of Tabuk, Saudi Arabia

NEW JERSEY

•

LONDON

12345_9789811239434_TP.indd 2

•

SINGAPORE

•

BEIJING

•

SHANGHAI

•

HONG KONG

•

TAIPEI

•

CHENNAI

•

TOKYO

24/1/22 9:57 AM

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Names: Cattani, Carlo, 1954– author. | Ben Mabrouk, Anouar, author. | Arfaoui, Sabrine, author. Title: Fractal analysis : basic concepts and applications / Carlo Cattani, University of Tuscia, Italy, Anouar Ben Mabrouk, University of Monastir, Tunisia & University of Kairouan, Tunisia, University of Tabuk, Saudi Arabia, Sabrine Arfaoui, University of Monastir, Tunisia, University of Tabuk, Saudi Arabia. Description: New Jersey : World Scientific, [2022] | Series: Series on advances in mathematics for applied sciences, 1793-0901 | Includes bibliographical references and index. Identifiers: LCCN 2021054614 | ISBN 9789811239434 (hardcover) | ISBN 9789811239441 (ebook) | ISBN 9789811239458 (ebook other) Subjects: LCSH: Fractal analysis. Classification: LCC QA614.86 .C38 2022 | DDC 514/.742--dc23/eng/20220104 LC record available at https://lccn.loc.gov/2021054614 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 2022 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/12345#t=suppl Desk Editors: Anthony Alexander/Lai Fun Kwong Typeset by Stallion Press Email: [email protected] Printed in Singapore

Alex - 12345 - Fractal Analysis.indd 4

5/1/2022 1:48:16 pm

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

To our mothers, fathers, and families, to our teachers and everyone who contributed to the success of this work

v

page v

B1948

Governing Asia

This page intentionally left blank

B1948_1-Aoki.indd 6

9/22/2014 4:24:57 PM

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Preface

Fractals may be observed almost everywhere in nature, and our daily life. They are observed in trees, rocks, snow, human body, cities, fractal smart grids . . . etc. This makes them attractive subjects in scientific researches in pure and applied fields. Fractals are also good models for the resolution of many problems in the pure mathematical point of view such as PDE, fractal domains and boundaries, dynamical systems, and turbulence modeling. In the applied point of view, fractals are applied in prices modeling, market indices, climate factors, very fine technology, smart cities, traffic, arts, design, nature, and bio domains. These facts have made them very interesting subjects in scientific researches, yielding a growing set of documentation about them. However, there still exists a necessity of more references to be developed, especially due to the restriction of the majority of books dealing with fractal analysis and geometry to specific community of readers. In the present volume, we have tried to elaborate a self-containing reference in the field of fractal analysis to be adapted for a large community of readers by providing the necessary developments. Further, in the present volume, basic concepts of measure theory as a main tool in fractal analysis will be recalled with the necessary developments. Besides, some concepts from stochastic calculus such as martingales will be provided with detailed proofs due to the complexity of such concepts. Next, the essential target of the book concerning the fractal dimensions, measures, sets are developed in a series of chapters, starting from the original definitions of Hausdorff measure and dimension, packing measures and dimension, the concept of capacity of sets and its relation to fractal

vii

page vii

February 7, 2022 16:4

viii

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Preface

measures and dimensions. Besides, the notion of multifractal formalism is also investigated with necessary developments, examples, and exercises. Finally, the volume is completed with some applications of the concepts exposed previously. To finish with this preface, we stress on the fact that the present volume, in fact, stems from many papers, lectures, presentations, and also discussions on the theory of fractal analysis, and its applications. The ideas gathered here are some times re-developed, improved, and completed with necessary developments. These may not be perfect and need sometimes to be criticized, corrected, and improved by readers. Any comments are welcome at any time. To participate in explaining and understanding the theory of fractal analysis, especially for beginners, some exercises and applications, that are simple to handle, are provided. We hope that the present volume provides a basic and self-contained introduction to the ideas underpinning fractal analysis, and its related fields, especially for master’s degree students, and PhD researchers. It may also serve for scientists and research in industrial sectors where the understanding and the construction of models for real world problems exhibits fractals. Recall that such concepts are now widely applied in finance, medicine, engineering, transport, images, signals, etc. This makes the present volume of interest for practitioners and theorists in these fields. We would like finally to thank all persons without whose help the present work could not be realized. We would like to thank World Scientific Publishing Co. for the opportunity to realize our project.

page viii

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

About the Authors

Carlo Cattani is currently Professor of Mathematical Physics and Applied Mathematics at the Engineering School (DEIM) of University of Tuscia. His scientific interests include, but are not limited to, wavelets, dynamical systems, fractals, fractional calculus, numerical methods, number theory, stochastic integro-differential equations, competition models, time-series analysis, nonlinear analysis, complexity of living systems, pattern analysis, computational biology, biophysics, history of science. He is the author (and co-author) of more than 300 scientific articles in international journals and many books. Anouar Ben Mabrouk is qualified as Professor of Mathematics, and Mathematics and Applications, from the French ministry of Higher Education. He is an Associate Professor of Mathematics at the University of Kairouan, Tunisia, and a member of the Laboratory of Algebra, Number Theory, and Nonlinear Analysis, Faculty of Sciences, University of Monastir, Tunisia, and the Department of Mathematics, Faculty of Sciences, Tabuk University, Saudi Arabia. His main focus is on wavelets, fractals, probability/statistics, PDEs and related fields such as financial mathematics, time series, image/signal processing, numerical and theoretical aspects of PDEs. Currently, he is Professor of Mathematics at the University of Tabuk, Saudi Arabia.

ix

page ix

February 7, 2022 16:4

x

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

About the Authors

Sabrine Arfaoui is Assistant Professor of Mathematics at the Laboratory of Algebra, Number Theory, and Nonlinear Analysis, University of Monastir, Faculty of Sciences, and the Department of Mathematics, Faculty of Sciences, Tabuk University, Saudi Arabia. Her main focus is on wavelet harmonic analysis, especially in the Clifford algebra/analysis framework and their applications in other fields such as fractals, PDEs, bio-signals/ bio-images. Currently, Dr. Arfaoui is attached to University of Tabuk, Saudi Arabia, in a technical cooperation project.

page x

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Contents

Preface

vii

About the Authors

ix

List of Figures

xv

List of Table

xvii

1.

Introduction

1

2.

Basics of Measure Theory

6

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 3.

σ-algebras . . . . . . . . . . . . . . . . . . Some topological concepts . . . . . . . . . Outer measures . . . . . . . . . . . . . . . Regular outer measures . . . . . . . . . . Metric outer measures . . . . . . . . . . . Lebesgue measure on Rd . . . . . . . . . . Convergence of measures on metric spaces Exercises for Chapter 2 . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

Martingales with Discrete Time 3.1 3.2 3.3 3.4

6 8 9 10 11 13 14 23 27

Generalities . . . . . . . . . . . . . . . . . Conditional expectation . . . . . . . . . . Convergence and regularity of martingales Regularity of integrable martingales . . .

xi

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

27 31 33 35

page xi

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

xii

Contents

3.5

3.6 4.

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

Hausdorff measure . . . . . . . . . . . . . . . . . . . Hausdorff dimension of Cantor-type sets . . . . . . . Other variants of Hausdorff dimension . . . . . . . . Upper and lower bounds of the Hausdorff dimension Frostman’s Lemma . . . . . . . . . . . . . . . . . . . Application . . . . . . . . . . . . . . . . . . . . . . . Exercises for Chapter 4 . . . . . . . . . . . . . . . . .

Generalities . . . . . Self-similar sets . . . Billingsley dimension Eggleston theorem . Exercises for Chapter

. . . . . . .

. . . . . . .

. . . . . . .

8.2 8.3 8.4

50 57 63 66 68 72 74 77

. . . . . . . . 5.

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

Bouligand–Minkowski dimension Packing measure . . . . . . . . . Packing dimension . . . . . . . . Exercises for Chapter 6 . . . . . .

77 80 91 95 97 100

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

101 104 110 113 116

The multifractal formalism . . . . . . . . . . . . . . . . . . 118 Existence of Gibbs measures . . . . . . . . . . . . . . . . . 123 Exercises for Chapter 7 . . . . . . . . . . . . . . . . . . . . 129

Extensions to Multifractal Cases 8.1

38 38 41 47 50

Multifractal Analysis of Gibbs Type Measures 7.1 7.2 7.3

8.

. . . .

Packing Measure and Dimension 6.1 6.2 6.3 6.4

7.

. . . .

Capacity Dimension of Sets 5.1 5.2 5.3 5.4 5.5

6.

Positive and upper martingales . . 3.5.1 Stopping time . . . . . . . 3.5.2 Positive upper martingales Exercises for Chapter 3 . . . . . . .

Hausdorff Measure and Dimension 4.1 4.2 4.3 4.4 4.5 4.6 4.7

5.

b4547-main

Generalized multifractal versions of the Hausdorff, packing measures, and dimensions . . . . . . . . . . Generalized Bouligand–Minkowski dimension . . . . The multifractal spectrum . . . . . . . . . . . . . . . Exercises for Chapter 8 . . . . . . . . . . . . . . . . .

134 and . . . . . . . . . . . .

135 149 159 174

page xii

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Contents

9.

xiii

Some Applications 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8

Introduction . . . . . . . . . . . . . . . . . . . . . . . Fractals in plants’ nature . . . . . . . . . . . . . . . Fractals in human body anatomy . . . . . . . . . . . Fractals for time series . . . . . . . . . . . . . . . . . Fractals for signals/images: The case of nano images A classical fractal self-similar set . . . . . . . . . . . A case of self-similar type measures . . . . . . . . . . Exercises for Chapter 9 . . . . . . . . . . . . . . . . .

179 . . . . . . . .

. . . . . . . .

. . . . . . . .

179 180 183 186 189 194 196 199

Bibliography

203

Index

215

page xiii

B1948

Governing Asia

This page intentionally left blank

B1948_1-Aoki.indd 6

9/22/2014 4:24:57 PM

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

List of Figures

4.1 4.2

The Hausdorff dimension of a set E. . . . . . . . . . . . . . . . The triadic Cantor set. . . . . . . . . . . . . . . . . . . . . . . .

6.1

The packing dimension of a set E. . . . . . . . . . . . . . . . . . 110

9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12 9.13 9.14 9.15

A natural Barnsley fern. . . . . . . . . . . . . . . . . . . . A numerical Barnsley fern. . . . . . . . . . . . . . . . . . . Human lung. . . . . . . . . . . . . . . . . . . . . . . . . . First step for 2-dimensional Mandelbrot model. . . . . . . The tree structure of human lung. . . . . . . . . . . . . . Stock index. . . . . . . . . . . . . . . . . . . . . . . . . . . Heart beat records. . . . . . . . . . . . . . . . . . . . . . . The series x(t) due to (9.1). . . . . . . . . . . . . . . . . . The scaling function of the series x(t) due to (9.1). . . . . The multifractal spectrum of the series x(t) due to (9.1). . The box dimension estimation of the Great Britain coast. The box dimension estimation of a black-white image. . . . Titanium dioxide TiO2 nanoparticles. . . . . . . . . . . . . The first 6 steps of the construction of Sierpinski carpet. . The first five iterations of a fractal antenna. . . . . . . . . .

xv

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

55 58 181 183 183 184 185 187 187 188 190 190 192 192 193 195 200

page xv

B1948

Governing Asia

This page intentionally left blank

B1948_1-Aoki.indd 6

9/22/2014 4:24:57 PM

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

List of Table

9.1

Estimation of the fractal dimension of TiO2 illustrated in Figure 9.13. . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

xvii

page xvii

B1948

Governing Asia

This page intentionally left blank

B1948_1-Aoki.indd 6

9/22/2014 4:24:57 PM

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Chapter 1

Introduction

Fractals in analysis, geometry, and generally in science are nowadays very popular concepts. The appearance of such concepts in science is due essentially to Mandelbrot by pioneering their application in scientific modeling starting from financial time series issued from cotton price to a wide variety of applications in turbulence, plants, coastlines, galaxies, etc. Indeed, in pure mathematics, fractals appeared in many contexts, such as the resolution of PDEs on fractal domains and PDEs with fractal boundaries ([Alimohammady et al (2017); Baleanu et al (2014); Cattani (2020); Xiaojun et al (2013); Xu et al (2014); Yang, Cattani et al (2014); Yang, Cattani and Xie (2015); Yang, Machado et al (2017); Yang, Srivastava et al (2015); Yang, Zhang et al (2014a,b); Yan et al (2014); Zhang, Cattani and Yang (2015)]). They may be also met in dynamical systems [Gervais (2009); Pesin (1997); Pesin and Climenhaga (2009)], and turbulence modeling [Benzi et al (1984); Cattani and Pierro (2013); Cattani (2010b); Frisch and Parisi (1985); Mandelbrot (1974)]. Since their appearance, fractals are also applied to explain the movement of prices, markets, financial indices, [Azizieh (2002); Benaych-Georges (2009); Ben Mabrouk, Ben Abdallah and Hamrita (2011); Calvet and Fisher (2008); Fan et al (2019); Fernandez-Martinez et al (2019); Fillol (2005); Hudson and Mandelbrot (2005); Mandelbrot (1997); Mandelbrot and Hudson (2006); Walter (2001)]. Besides, fractals are also applied for modeling instruments, such as fractal antenna ([Anguera et al (2020)], climate factors, and geophysical targets [Bozkus et al (2020); Chu (1999); Figueiredo et al (2014); Scholz and Mandelbrot (1989)]. In bio domains, such as medical applications, fractals are nowadays famous models [Badea et al, 2013; Castiglioni-Faini, 2019; Cattani, Pierro and Altieri, 2012; Karaca and Cattani, 2017; Mauroy et al, 2004; Sapoval and Filoche, 2010; Weibel, 1963].

1

page 1

February 7, 2022 16:4

2

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Finally, we may also find them in signal/image processing, such as [Barnsley et al (1988); Cattani and Ciancio (2016); Chen, Cattani and Zhong (2014); Li et al (2014); Liu et al (2019); Vehel and Legrand (2004); Vehel and Vojak (1998)], traffic, arts, design, nature, etc. [Briggs (1992); Flake (1998); Li, Zhao and Cattani (2013); Mandelbrot (1982); Novak (2004); Pickover (1995); Scheuring and Riedi (1994); Wang et al (2014)]. Fractals have emerged in pure mathematics and pure mathematical analysis, and geometry has been developed as part and/or extension of measure theory, dimension theory, dynamical systems, and also nonEuclidean geometry. Mathematically, a Fractal set according to Mandelbrot derived its definition from the latin word fractus, which means in English ‘broken’, which explains the idea of characterizing next fractals by means of their Hausdorff dimension; generally and/or preferably, a non-integer real number quantity, which needs to be different from the topological dimension. Many variants have been developed by mathematicians to investigate such a notion. See the big list of references [Aversa and Bandt (1990); Baker and Schmidt (1970); Baker (1976); Barlow and Taylor (1992); Barral et al (2003); Batakis and Heurteaux (2002); B´elair (1987); Ben Nasr (1994, 1997); Ben Nasr and Bhouri (1997); Bernay (1975); Billingsley (1960, 1961); Billingsley and Henningsen (1975); Buck (1970, 1973); Cajar (1981); Cawley and Mauldin (1992); Cole (2000); Cole and Olsen (2003); Colebrook (1970); Collet et al (1987); Dai (1995); Dai and Liu (2008); David and Semmes (1997); Debussche (1998); Dryakhlov and Tempelman (2001); Edgar (1998, 2008); Eggleston (1949); Falconer (1994, 1985, 1990); Gurevich and Tempelman (1999); Heurteaux (1998); Lucas (2000); Makarov (1985); Mandelbrot (1993, 1995); Mattila (1995); Ngai (1997); Olivier (1998); Pesin (1997); Pesin and Climenhaga (2009); Riedi and Scheuring (1997); Rogers (1970); Selezneff (2011); Spear (1992); Taylor (1995); Veerman (1998); Wu (1998, 2005); Zeng et al (2012); Zhou and Feng (2011); Zhu and Zhou (2014)]. One of the simple ideas ’to understand’ the concept of fractal dimensions, especially for non mathematicians, is to compare with the classical concepts of size (or the Lebesgue measure) when enforcing a scale change to the object in hand. For example, given an interval, or a segment I = [a, b], its Lebesgue measure (length) is obviously `(I) = b − a. Multiplying I by a scale λ ∈ R∗ , for example, results in an interval λI whose Lebesgue measure is `(λI) = |λ|(b − a) = |λ|1 `(I). In 2-dimensional Euclidean space, a rectangle I = [a, b] × [c, d] has the Lebesgue measure (surface) `(I) = (b − a)(d − c). Multiplying I by a scale

page 2

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Introduction

b4547-main

3

λ ∈ R∗ , as previously, results in a rectangle λI whom Lebesgue measure is `(λI) = |λ|2 (b − a)(d − c) = |λ|2 `(I). Qn Generally, in the n-dimensional Euclidean space, a cube I = i=1 [ai , bi ] Qn has the Lebesgue measure `(I) = i=1 (bi − ai ). Multiplying I by a scale λ ∈ R∗ results in a cube λI whom Lebesgue measure is `(λI) = |λ|n

n Y

(bi − ai ) = |λ|n `(I).

i=1

The notion of dimension in the fractal analysis, and geometry aims to generalize this concept to general variants of dimensions by seeking suitable measures µ, which associates to the set λI an analogue quantity |λ|s µ(I) for some generally non-integer s. This number will be the fractal dimension of the set I. Many forms of measures have been introduced in the mathematical/physical literature to investigate this notion, such as, the Hausdorff, Caratheodory, packing, . . . , etc. Each of the introduced measures has been related to a notion of dimension, which describes the aim above. These measures, and dimensions have been next applied to re-define the concept of fractal sets, by imposing some equality on the different dimensions, and formalisms. The majority of formalisms are valid for Cantor type sets. In the general case, the concept of being Fractal or not is still under study, and needs more considerations. One of the main problems behind the ambiguities is the fact that the computation of the fractal dimensions introduced, such as, the Hausdorff one is its impossibility, in the majority of cases, to be evaluated directly from the mathematical definition. We always have to come back to comparisons with Cantor types, self-similar, and scaling law. In some cases, we need to pass from the set to functional theory, and study instead of a set an associated function, which is generally irregular, and vice-versa. These facts, and difficulties are the main drawbacks, especially, in the applied fields when fractal models are investigated. Indeed, from the applied point of view, nowadays, fractals may be observed almost everywhere in nature, and our daily life. They are observed in trees, rocks, snow, human body, cities, Fractal smart grids . . . etc. This makes them attractive subjects in scientific researches in pure, and applied fields.

page 3

February 7, 2022 16:4

4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Since their introduction in the early works of Mandelbrot, the documentation on fractals and their applications have been growing widely. Although, with the rapid developments in technology, and the appearance of many new natural phenomena, there still exists a great need to understand such new problems and their modeling with fractals. The majority of books dealing with fractal analysis and geometry are somehow restricted to specific readers, and, in the majority, addressed to a pure mathematical community. This is comprehensive, and may be explained by the need of the developments of the original, and the first theoretical steps, and basic concepts for the theory. Next, with the dramatic development in technology, and science, fractal analysis has become a need in academic studies, especially, for young researchers. Consequently, the scientific academic community has had a great need to develop references in different forms, such as self-containing references, which respond mostly to the questions of young readers in Master’s and PhD levels. The present volume will be part of these selfcontaining references, and will help readers to win a lot of time by providing the necessary developments. This book is composed of nine chapters. A first introductory one in which a literal introduction is developed discussing the topic generally. Chapter 2 is concerned with the presentation of the basic concepts of measure theory, which constitutes the main tool and the main prerequisite for fractal analysis. Original developments of measure theory are provided with the necessary details. Chapter 3 is concerned with the study of the notion of martingales, especially in the discrete case. This chapter constitutes a part of the useful concepts applied in fractal analysis, and thus responds to our aim to provide a self-contained reference in fractal analysis. The essential results to be applied in the nest chapters are recalled with necessary details. Chapter 4 constitutes the starting step in the target theory to be developed in the present volume. Indeed, this chapter is the basic tool and the first step in fractal analysis and fractal geometry, as well as any field applying these notions. Basic construction of the so-called Hausdorff measure and the associated Hausdorff dimension will be exposed in detail. Chapter 5 is devoted to the presentation of another variant of fractal dimensions, the so-called capacity dimension of sets, which coincides in some cases with the Hausdorff dimension. In Chapter 6, the notion of the packing measure and packing dimension are exposed as other variants of fractal measures and dimensions.

page 4

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Introduction

b4547-main

5

Chapter 7 is devoted to the presentation of the multifractal formalism for measures. In this chapter, we essentially focus on the simple case of multifractal measures where the construction of associated Gibbs ones is always possible. It is shown that the possibility of constructing Gibbs measures on the supports of the multifractal measures is the essential cause of the validity of the multifractal formalism. The first developments in the same direction of the subject presented in this chapter are due to [Brown et al (1992); Peyriere (1992)]. Besides, for more details, and more examples, and cases, readers may refer also to [Ben Mabrouk (2008a,b,c); Ben Mabrouk and Aouidi (2011, 2012); Ben Mabrouk, Aouidi and Ben Slimane (2014, 2016); Ben Nasr (1994, 1997); Ben Nasr and Bhouri (1997); Ben Nasr, Bhouri and Heurteaux (2002); Billingsley (1960, 1961, 1965, 1968); Billingsley and Henningsen (1975); Brown et al (1992); David and Semmes (1997); Edgar (1998, 2008); Falconer (1994, 1990); Heurteaux (1998); Mandelbrot (1995); Menceur, Ben Mabrouk and Betina (2016); Menceur and Ben Mabrouk (2019); Mignot (1998); Ngai (1997); Olsen (1995); Pesin (1997); Pesin and Climenhaga (2009); Peyriere (1992); Riedi (1995); Rogers (1970); Spear (1992); Wu (1998)]. Chapter 8 is devoted to the presentation of the multifractal formalism for measures. It is concerned with the exposition of the multifractal extensions of the notions exposed in the previous chapters. Multifractal generalizations of Hausdorff and packing measures as well as their associated dimensions are exposed. We recall in this context that this chapter is essentially based on Olsen’s works in [Cole and Olsen (2003); Cole (2000); Olsen (1995, 1996, 2000); O’Neil (1997)]. However, more information, developments, and extensions may also be found in [Ben Mabrouk (2008a,b,c); Ben Mabrouk and Aouidi (2011, 2012); Ben Mabrouk, Aouidi and Ben Slimane (2014, 2016); Ben Nasr (1994, 1997); Ben Nasr and Bhouri (1997); Ben Nasr, Bhouri and Heurteaux (2002); Menceur, Ben Mabrouk and Betina (2016); Menceur and Ben Mabrouk (2019)]. Finally, Chapter 9 is concerned with the development of some applications of the concepts exposed in the previous chapters, in order to show the utility of the notion of the fractal analysis, on the one hand, and to provide the community of non mathematicians with some ideas of concrete applications, on the other.

page 5

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Chapter 2

Basics of Measure Theory

We propose in this chapter to recall some basic concepts of measure theory that will be applied later. We will try to review with some details all the concepts needed from measure theory, and focus on the most closed concepts to fractal analysis, and geometry, which constitute the principal goal of the book. Readers may refer to [David and Semmes (1997); Edgar (1998, 2008); Falconer (1990); Lambert (2012); Robert (2020); Yeh (2014)]. 2.1

σ-algebras

Along the present chapter, B will designate a σ-algebra on the Euclidean space Rd , d ∈ N is an integer. Definition 2.1. Let X be a set, and F be a collection of subsets of X. F is said to be a σ-algebra on X if it satisfies the following assertions. i. ∅ ∈ F. ii. For any element A ∈ F, its complement Ac ∈ F. iii. For any countable collection (An )n ⊂ F, the union An ∈ F. n

For any set X there are always two natural examples of σ-algebras on it. • The one composed of {∅, X} known as the coarse σ-algebra. • The one composed of all subsets of X (the power set of X) denoted P(X). • Consequently, any σ-algebra lies between the two types above. As a consequence of Definition 2.1, the σ-algebra may be seen otherwise. Proposition 2.1. Let X be a set, and F be a σ-algebra on X. The following assertions are true. 6

page 6

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Basics of Measure Theory

(1) (2) (3) (4) (5)

b4547-main

7

The whole set X is an element of F, (X ∈ F). F is invariant under finite unions. F is invariant under finite intersections. F is invariant under countable unions. F is invariant under the difference of sets.

Proof. 1. We know, from Assertion (i) Definition 2.1 above, that ∅ ∈ F. Hence, by Assertion (ii) in Definition 2.1, its complement ∅c ∈ F. As ∅c = X, we get X ∈ F. 2. Let A0 , A1 , . . . , An , (n ∈ N fixed) be a finite collection of elements of F, and consider the countable collection (Bi )i∈N , such that, Bi = Ai , for 1 ≤ i ≤ n, and Bi = ∅, for i ≥ n + 1. It is straightforward that (Bi )i∈N ⊂ F. Hence, Assertion (iii) in Definition 2.1 yields that the union Bi ∈ F. Now, observe that i∈N

[

B i = A0 ∪ A1 ∪ · · · ∪ An .

i∈N

It results that A0 ∪ A1 ∪ · · · ∪ An ∈ F. 3. Let A0 , A1 , . . . , An , (n ∈ N fixed) be a finite collection of elements of F, and consider the collection B0 , B1 , . . . , Bn , such that Bi = Aci , for 1 ≤ i ≤ n. By Assertion (ii) in Definition 2.1, we conclude that B0 , B1 , . . . , Bn are elements of F. Assertion 2 above yields that their union B0 ∪ B1 ∪ · · · ∪ Bn ∈ F. Now, using again Assertion (ii) in Definition 2.1, we obtain  c B0 ∪ B1 ∪ · · · ∪ Bn ∈ F. Observe next that  c B0 ∪ B1 ∪ · · · ∪ Bn = A0 ∩ A1 ∩ · · · ∩ An . We thus get A0 ∩ A1 ∩ · · · ∩ An ∈ F. 4. Let A0 , A1 , . . . , An , . . . be countable collection of elements An ∈ F, and consider the countable collection (Bn )n∈N , such that, Bn = Acn , ∀n. We

page 7

February 7, 2022 16:4

ws-book9x6

8

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

thus obtain from Assertion (ii), Definition 2.1, that (Bn )n∈N is a countable collection of elements of F. Hence, Assertion (iii) in Definition 2.1 yields that the union ∪ Bn ∈ F. Now, observe that n∈N  c ∪ B n = ∩ An . n∈N

n∈N

It results, from Assertion (ii) in Definition 2.1 again, that ∩ An ∈ F. n∈N

5. This is an easy consequence of the previous assertions. Indeed, let A, B ∈ F. Then, A, B c ∈ F. It follows, from Assertion 3 above, that A ∩ B c ∈ F, which is equivalent to A \ B ∈ F.  2.2

Some topological concepts

In this subsection, we aim to recall some basic concepts from topology, which will be used later. Definition 2.2. A topology on a set X is a subset T of P(X) satisfying i. ∅, and X are elements of T . ii. T is invariant under arbitrary unions. iii. T is invariant under finite intersections. The pair (X, T ) is said to be a topological space. The elements of T are called open sets in X. A subset of X is said to be closed if its complementary is open. As for the case of σ-algebras, for any set X, there are always two natural examples of topology on it. • The topology T = {∅, X} known as the trivial or the coarse topology. • The topology T = P(X) known as the discrete topology. Example (1) X = {1, 2, 3}. T = {∅, {1}, {2}, {1, 2}, X} is a topology. (2) X = {1, 2, 3}, and T = {∅, {1}, {2}, {1, 2}, X}. T is not a topology. (3) X = N, and T = {A ⊂ N, A is finite} ∪ {N }. T is not a topology, because (for example) the union of all finite sets, not containing zero, is infinite but is not in T . One of the important cases of topological spaces are those associated to metrics known as metric spaces.

page 8

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Basics of Measure Theory

b4547-main

9

Definition 2.3. We call metric on a non-empty set X, any mapping d : X × X −→ [0, ∞[, satisfying i. d(x, y) = 0 ⇐⇒ x = y, ∀x, y ∈ X. ii. d(x, y) = d(y, x), ∀x, y ∈ X. iii. d(x, y) ≤ d(x, z) + d(z, y), ∀x, y, z ∈ X. The couple (X, d) is called a metric space. Definition 2.4. Let (X, T ) be a topological space. We call a Borel σalgebra the smallest σ-algebra containing the topology T . The pair (X, B) is called a measurable space. The following concepts are known in measure theory. Let (X, B) be a measurable space, and µ : B → [0 + ∞] be a positive measure on X. • If µ(X) < +∞, µ is said to be finite or bounded measure. • µ(X) is called the total mass of µ. • If µ(X) = 1, µ is called a probability measure. 2.3

Outer measures

The concept of outer measures is a refinement of the notion of measures, in the sense that, it excludes some characteristics. We have precisely the following definition. Definition 2.5. An outer measure on a set X is any map µ : P(X) → [0, +∞], satisfying the following assertions. • µ(∅) = 0. • µ(A) [ ≤ µ(B), Xfor all A, B ⊂ X, such that, A ⊂ B. • µ( An ) ≤ µ(An ), for any sequence (An )n in P(X). n

n

Definition 2.6. Let µ be an outer measure on X. A subset A ⊂ X is said to be µ-measurable if µ(E) = µ(E ∩ A) + µ(E ∩ Ac ), ∀E ⊂ X. Theorem 2.1. Let µ be an outer measure on X. The collection B of all µ-measurable of X is a σ-algebra on X. Moreover, the restriction µB of µ on B is a measure.

page 9

February 7, 2022 16:4

ws-book9x6

10

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Proof. i. We shall show that µ(E) = µ(E ∩ X) + µ(E ∩ X c ), ∀E ⊂ X, and µ(E) = µ(E ∩ ∅) + µ(E ∩ ∅c ), ∀E ⊂ X, which is true as E ∩ X = E ∩ ∅c = E

and E ∩ ∅ = E ∩ X = ∅.

ii. For all A ∈ B, we have µ(E) = µ(E ∩ A) + µ(E ∩ Ac ), ∀E ⊂ X, which is equivalent to µ(E) = µ(E ∩ Ac ) + µ(E ∩ (Ac )c ), ∀E ⊂ X. Consequently, Ac ∈ B. iii. Let (An )n≥0 be a sequence of elements of B. Denote A = ∪n An and Yn = ∪nk=0 Ak . For each n ≥ 0, the set Yn is µ-measurable, as finite union of µ-measurable. Moreover, the sequence (Yn )n is increasing to A. So, for any subset E ⊂ X, we have for any subset E ⊂ X, µ(E) = µ(E ∩ Yn ) + µ(E ∩ Ync ) ≥ µ(E ∩ Yn ) + µ(E ∩ Ac ). Observe now that the sequence (Yn )n is increasing to A. It results that (µ(E ∩ Yn ))n is increasing to µ(E ∩ A). This completes the proof. 2.4

Regular outer measures

Definition 2.7. An outer measure µ on X is called regular if ∀ E ⊂ X, there exists a µ-measurable set A containing E, and satisfying µ(E) = µ(A). Lemma 2.1. Let µ be an outer regular measure on X, and (En )n be an increasing sequence of elements of P(X), then µ( lim En ) = lim µ(En ). n→+∞

n→+∞

page 10

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Basics of Measure Theory

11

Proof. As the sequence (En )n is increasing, it holds that µ( lim En ) ≥ lim µ(En ). n→+∞

n→+∞

Now, denote B the σ-algebra of µ-measurable subsets in X. µ being regular, hence, ∀n ≥ 0, there exists An ⊃ En , An ⊂ B, such that, µ(An ) = µ(En ). We have lim En = lim inf En .

n→∞

n

Consequently, µ( lim En ) = µ(lim inf En ) ≤ µ(lim inf (An )). n→∞

n

n

This yields, using Fatou’s Lemma, that µ( lim En ) ≤ lim inf µ(An ) n→∞

n

= lim inf µ(En ) n→∞

= lim µ(En ). n→∞

2.5

Metric outer measures

Let (X, d) be a metric space. Definition 2.8. An outer measure µ on X is called metric iff µ(E ∪ F ) = µ(E) + µ(F ), ∀E, F ⊂ X satisfying d(E, F ) > 0. Proposition 2.2. Let µ be a outer measure on (X, d), (An )n ⊂ X an increasing sequence, and A = ∪n An = lim An . n−→∞

Assume further that d(An , AAn+1 ) > 0, ∀n. Then, µ(A) = lim µ(An ). n−→∞

page 11

February 7, 2022 16:4

ws-book9x6

12

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Proof. Denote for k ∈ N, Ck = Ak+1 \ Ak the annulus between Ak+1 , and Ak . We immediately get d(Cj , Cp ) > 0, ∀ p ≥ j + 2, and ∞ [

A = An ∪ (

Ck ), ∀ n.

k=n+1

Consequently, µ(A) ≤ µ(An ) +

∞ X

µ(Ak+1 r Ak )

n+1

= µ(An ) + Rn , where we denoted for n ∈ N, Rn =

∞ X

µ(Ak+1 r Ak ).

n+1

We shall show that Rn −→ 0 as n −→ ∞. It suffices to prove that the series P P k≥1 µ(C2k ), and k≥1 µ(C2k+1 ) are convergent. Indeed, n X

µ(C2k+1 ) = µ(

n [

C2k+1 )

k=1

k=1

≤ µ(A2n+2 ) < µ(A). Similarly, we do for

X

µ(C2k ). As a result,

k≥1

µ(A) ≤ lim µ(An ). n→∞

Theorem 2.2. Let µ be an outer metric measure on (X, d). Then, every Borel subset of X is µ-measurable. Proof. Let B be a Borel subset of X, and denote for n ≥ 1, n 1o Bn = x ∈ EB such that d(x, B) ≥ . n

page 12

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Basics of Measure Theory

13

It is straightforward that B is a closed, and E ⊂ X. Moreover, (Bn )n is an increasing sequence. It satisfies also [ Bn = E \ B, n

and d(Bn , (E \ B) \ Bn+1 ) > 0, ∀ n. It follows, from the Proposition 2.2 above, that µ(E) ≥ µ(Bn ∪ (E ∩ B)) = µ(Bn ) + µ(E ∩ B). Letting n −→ ∞, we obtain µ(E) ≥ µ(E \ B) + µ(E ∩ B).

2.6

Lebesgue measure on Rd

We designate by P = (a1 , b1 ) × · · · × (ad , bd ), (ai < bi real numbers), a cube of order d in Rd , and denote V (P ) =

d Y

(bi − ai )

i=1

its volume. The Lebesgue measure on Rd is defined for E ⊂ Rd by X m(E) = inf V (Pj )/(Pj )j , where the lower bound is taken on the set of all cubes (Pj )j covering E (E ⊂ ∪j Pj ). It is straightforward that (1) (2) (3) (4)

m is an outer metric measure on Rd called Lebesgue outer measure. m(P ) = V (P ) for all cube P in Rd . Borel sets are obviously Lebesgue measurable. m is a regular outer measure on BRd .

Definition 2.9. The restriction of m on the σ-algebra of m-measurable subsets in Rd is called the Lebesgue measure in dimension n.

page 13

February 7, 2022 16:4

ws-book9x6

14

2.7

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Convergence of measures on metric spaces

Let (X, d) be a metric space, and B the Borel σ-algebra on X. Let also µ be a measure on B. Recall that µ is said to be finite if µ(X) < ∞. In particular, µ is said to be a probability measure on X if µ(X) = 1. We recall finally that an element E ∈ B is said to be regular relatively to the finite measure µ if it satisfies the assumption µ(E) = sup{µ(K), K is closed, K ⊂ E} = inf{µ(O), O is open, O ⊃ E}. The measure µ is said to be regular if any element E ∈ B is regular. We immediately have a preliminary result on regular measures. Lemma 2.2. Let µ be a regular measure on a measurable space (X, B). A set E ∈ B is µ-regular if, and only if, ∀ > 0, there exists an open set G, and a closed set F satisfying, F ⊂ E ⊂ G, and µ(G \ F ) < . Proof. =⇒) E ∈ B is µ-regular yields that ∀ > 0, there exists a closed set F ⊂ E, such that  µ(E) − < µ(F ) ≤ µ(E), 2 and  µ(E) ≤ µ(G ) < µ(E) + . 2 We immediately obtain   µ(E) − < µ(F ) ≤ µ(E) ≤ µ(G ) < µ(E) + . 2 2 As a result,   µ(G \F ) = µ(G ) − µ(F ) < + =  2 2 ⇐=) We observe that E\F ⊂ G\F . Hence, µ(E\F ) ≤ µ(G\F ) < , which yields that µ(F ) ≤ µ(E) < µ(F ) + , or equivalently µ(E) −  < µ(F ) ≤ µ(E). Hence, µ(E) = sup µ(F ). F ⊆E

page 14

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Basics of Measure Theory

15

Now, observe that similarly that G\E ⊂ G\F . Consequently µ(G\E) < , which yields that µ(E) ≤ µ(G) < µ(E) + . As a result, µ(E) = inf µ(G). G⊃E

As a result of this lemma, we obtain the following result. Proposition 2.3. Every finite measure µ on B is regular. Proof. Denote T = {E ∈ B, E is µ-regular}. We shall show that T = B. The inclusion T ⊂ B is obvious. Furthermore, ∅, X ∈ T . The collection T is also invariant by passage to the complement. We will show now that it S is invariant under countable unions. So, let (En )n ⊂ T , denote E = En , n

and let  > 0. It holds immediately that ∀n, there exists an open set On, , and a closed set Fn, such that  Fn, ⊂ En ⊂ On, and µ(On, \ Fn, ) < n . 3 S Let next O = On, . It consists of an open set that contains E. Let also n S F = Fn, . As µ is finite, there exists n0 ∈ N such that, n

µ(F \

n0 [ n=1

Consider the set F =

n0 [

Fn, )
0.

page 15

February 7, 2022 16:4

16

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Then, F is regular. It result that T is a σ-algebra, which contains the closed sets. Consequently, T ⊃ B. As a result, T = B. Definition 2.10. Let C0 be the set of bounded continuous functions on X, and (µn )n be a sequence of finite measures on B. We say that (µn )n converges weakly to a measure µ, and we write µn =⇒ µ, iff Z Z gdµn −→ gdµ, ∀g ∈ C0 , as n −→ ∞. X

X

Definition 2.11. A set E is said to be a continuity set for µ if µ(∂E) = 0. We immediately have the following theorem. Theorem 2.3. Let (µn )n be a sequence of finite measures on B, and µ be a finite measure on B also. The following assertions are equivalent. (1) µn =⇒ µ. (2) µ(C) ≥ lim µn (C), for all closed set C in X. n→∞

(3) µ(O) ≤ lim µn (O), for all open set O in X. n→∞

(4) lim µn (E) = µ(E), for all continuity set E of µ. n→∞

Proof. (i ⇒ ii) Let C be a closed subset in X, and for p ∈ N, denote n 1o Sp = x ∈ X, d(x, C) < p . 2 Hence, C, and Spc are disjoint closed subsets in X. Consequently, there exists a function gp ∈ C0 , such that, 0 ≤ gp ≤ 1, gp ≡ 1 on C, and gp ≡ 0 on Spc . Now, as (Sp )p decreases to its limit C, we obtain Z χc dµn lim µn (C) = lim n→∞ n→∞ X Z ≤ lim gp dµn n→∞ X Z = gp dµ. X

Next, as gp ∈ C0 , it follows, ∀ p ∈ N, that Z lim µn (C) ≤ gp dµ n→∞ ZX = gp dµ Sp

≤ µ(Sp ).

page 16

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Basics of Measure Theory

17

Letting p → +∞, we get lim µn (C) ≤ µ(C).

n→∞

(ii ⇒ iii) It is easy by considering the complementary. (iii ⇒ iv) Let E be such that µ(∂E) = 0. So, ◦

µ(E) = µ(E) = µ(E). As E is closed, we obtain lim µn (E) ≤ lim µn (E) ≤ µ(E) = µ(E).

n→∞

n→∞

◦

Similarly, as E is open, we get ◦

◦

µ(E) = µ(E) ≤ lim µn (E) ≤ lim µn (E). n→∞

n→∞

(iv ⇒ i) Fix  > 0, let g ∈ C0 , and µg be defined on the Borel σ-algebra BR on R by µg (A) = µ{x ∈ X/g(x) ∈ A}, ∀ A ∈ BR . It is easy to see that µg is a finite measure on BR . Moreover, as g is bounded, there exists an interval [a, b] ⊂ R, such that, µg ≡ 0 on [a, b]c . Consider next a subdivision t0 = a < t1 < · · · < tk = b, such that tj+1 −tj < , µg {tj } = 0, ∀ j = 1, 2, . . . , k, and let Ej = {x ∈ X, tj−1 < g(x) ≤ tj }. S The Ej ’s are disjoint Borel sets. Moreover, X = j Ej . Now, observe that [ ∂Ej = {g(x) = tj−1 } {g(x) = tj }, we deduce that (∂Ej ) = 0. Consequently, ∀ j = 1, 2, . . . , k we obtain µn (Ej ) → µ(Ej ) as n → ∞. Denote next h =

Pk

j=1 tj χEj .

We get

sup |g(x) − h(x)| < . x∈X

Consequently, for n ∈ N, we obtain Z Z Z | gdµn − gdµ| ≤ |g − h|dµn X X XZ Z +| hdµn − hdµ| ZX X + |g − h|dµ X

≤ µn (X) + µ(X) k X + |tj ||µn (Ej ) − µ(Ej )|. j=1

This completes the proof.

page 17

February 7, 2022 16:4

18

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Corollary 2.1. Let µ, ν be finite measures on X, such that, Z Z gdµ = gdν X

X

for any continuous function g ∈ C0 . Then, µ ≡ ν. Proof. It follows from Theorem 2.3. Indeed, consider the sequence (µn )n , such that, µn = ν. We immediately observe that µn ⇒ µ. Hence, for all closed set F in X, we have ν(F ) ≤ µ(F ). This means that µ ≡ ν on the closed sets in X. Next, as µ, and ν are regular, it follows that µ ≡ ν on B. Corollary 2.2. Let (µ)n be a sequence of finite measures on B, µ, and ν be finite measures on B also, such that, µn ⇒ µ, and µn ⇒ ν. Then, µ ≡ ν. Proof. It follows from Corollary 2.1 immediately. Indeed, ∀g ∈ C0 , we get Z Z g dµn → g dµ, n→+∞

X

and

Z

X

Z g dµn

X

→

n→+∞

g dν. X

Hence, Z

Z g dν, ∀g.

g dµ = X

X

Therefore µ ≡ ν. Definition 2.12. Let M = {µt , t ∈ T } be a collection of finite measures on B. (1) M is said to be relatively compact, if any sequence (µn )n in M has a sub-sequence convergent weakly to a finite measure on B. (2) M is said to be tight, if for all  > 0, there exists a compact K ⊂ X, such that, sup µt (X \ K ) < . t∈τ

Example Consider X = R, and F = {ft , t ∈ T } a collection of bounded functions on R, which are decreasing, and right continuous,, and M the set of corresponding finite measures. The following theorem due to Prokorov relates the concept of relatively compact measures, and the tight ones. Theorem 2.4. Prokorov Theorem. Let (X, d) be a complete separable metric space. Let BX be the Borel σ-algebra on X, and M be a collection of probability measures on BX . Then, M is relatively compact if, and only if M is tight.

page 18

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Basics of Measure Theory

b4547-main

19

Proof. =⇒) Assume that M is relatively compact,, and let  > 0. We shall show that, there exists a compact K ⊂ X, such that µ(K ) > 1 − , ∀ µ ∈ M. Let k ∈ N. As X is separable, there exists {xn }n ⊂ X dense in X. Denote so 1 Sn,k = {x ∈ X, d(xn , x) < }. k [ We observe that Sn,k = X. Denote next for n ∈ N, n

Un,k =

n [

Sj,k .

j=1

It consists of a sequence of open sets increasing to X. Assume thus that there exists 0 , such that, ∀n, there exists a measure µn satisfying 0 µn (Un,k ) ≤ 1 − k . 2 As {µn } ⊂ M, there exists a sub-sequence µ(ϕ(n)) convergent weakly to µ ∈ M. We immediately get µ(Us,k ) ≤ lim µϕ(n) (Us,k ) n→∞

≤ lim µϕ(n) (Uϕ(n),k ) n→∞ 0 ≤ 1− k. 2 As a consequence, µ(Us,k ) ≤ 1 −

0 , ∀s. 2k

Letting s → ∞, we get 0 , 2k which contradicts the fact that µ(X) = 1. Therefore, there exists nk ≤ 1, such that, 0 µ(Unk ,k ) > 1 − k , ∀µ ∈ M. 2 So, denote µ(X) ≤ 1 −

K =

nk \ [ k≥1 j=1

Sj,k .

page 19

February 7, 2022 16:4

ws-book9x6

20

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

It is now straightforward that K is compact, and µ(K ) > 1 − , ∀µ ∈ M. ⇐=). Assume that M is tight. So, ∀n ≥ 0, there exists a compact Kn in X, such that, 1 µ(Kn ) > 1 − n . 2 Now, as X is separable, there exists a countable set of open balls U = {Sp , p ≥ 0}, which constitutes a neighborhoods’ basis of X. Denote so ( ) [ \ F= S Kn S ∈ U . finite It is easy to check that F is countable, invariant under finite union, and its elements are compact. Hence, for any sequence (µn )n ⊂ M, there exists a sub-sequence (µnk )n ⊂ (µn )n , for which, lim µnk (F ) exists, and finite for k→∞

all F ∈ F. Denote next ν(F ) = lim µnk (F ). k→+∞

Define now the measure µ on B by µ(O) =

sup

ν(F ),

(2.1)

F ⊂O, F ∈F

for all open set O in X. Observing that ν(F ) = lim µnk (F ) ≤ limk µnk (O), k→+∞

we deduce that µ(O) ≤ limk µnk (O). As a consequence, Theorem 2.3 yields that µnk =⇒ µ. So, it suffices to prove that ν defines well a probability measure µ on X. To do this, consider the mapping µ∗ defined on P(X) by µ∗ (E) =

inf

O⊃E, O

, µ(O), ∀E ∈ P(X). open

As ν(φ) = 0, we immediately get µ∗ (φ) = 0. Next, for E1 ⊂ E2 in P(X), let O be an open set containing E2 . We have µ∗ (E1 ) ≤ µ(O), which yields that µ∗ (E1 ) ≤ µ∗ (E2 ).

page 20

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Basics of Measure Theory

Let now (En )n be a countable set in P(X), and denote E =

b4547-main

21

[

En . Observe

n

next that, for all  > 0, and all n ∈ N, there exists open sets On , such that  µ(On ) ≤ µ∗ (En ) + n , 2 S which, with the fact that E ⊂ n On , yields that [ µ∗ (E) ≤ µ( On ). n

It suffices so to show that [ X µ( On ) ≤ µ(On ). n

(2.2)

n

S To do it, let F ∈ F be, such that, F ⊂ n On . As F is compact, there S n0 exists n0 ∈ N, such that, F ⊂ n=1 On , which, by the next, yields that, ν(F ) ≤ µ(

n0 [

On ).

n=1

Hence, it suffices to show that (2.2) holds for a set of two elements O1 , and O2 in F, such that, F ⊂ O1 ∪ O2 . Denote thus n o C1 = x ∈ F, d(x, O1c ) ≥ d(x, O2c ) , and n o C2 = x ∈ F, d(x, O2c ) ≥ d(x, O1c ) . It is easy to see that C1 , and C2 are compact, C1 ⊂ O1 , and C2 ⊂ O2 . On the other hand, there exists x1,j ..., xNj ,j ∈ C1 , such that Cj ⊂

Nj [

Nj [

Sxs ,j ⊂

s=1

S xs ,j ⊂ θj , 1 ≤ j ≤ 2.

s=1

So, as F ∈ F, there exists compact sets Knj , satisfying F ⊂ Knj , which implies that Cj ⊂

Nj [

S xs ,j

\

Knj ⊂ O1 .

s=1

Denote next Fj =

Nj [ s=1

S xs ,j

\

Knj .

page 21

February 7, 2022 16:4

ws-book9x6

22

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Hence, Fj ∈ F. Moreover, there exists F1 , F2 ∈ F, such that, C1 ⊂ F1 ⊂ O1 , C2 ⊂ F2 ⊂ O2 , and that F ⊂ C1

[

C2 ⊂ F1

[

F2 ⊂ O1

[

O2 .

As a result, ν(F ) ≤ ν(F1 ) + ν(F2 ) ≤ µ(O1 ) + µ(O2 ), which yields that [

O2 ) ≤ µ(O1 ) + µ(O2 ). [ Consequently, whenever F ⊂ On , we get µ(O1

n

ν(F ) ≤ µ(

n0 [

On ) ≤

n=1

n0 X

µ(Oj ) ≤

j=1

X

µ(Oj ).

j≥1

Hence, µ(

[ n

On ) ≤

X

µ(On ).

n

As a result, [ [ X [ µ? ( En ) ≤ µ( θn ) ≤ µ(θn ) ≤ µ? ( En ) + . n

n

n

We thus proved that µ? is an outer measure. It remains to show that the σ-algebra B ? = {E ∈ B, E is µ? -measurable} = B. So, let A ∈ P(X). We have µ? (E) ≤ µ? (E ∩ A) + µ? (E ∩ Ac ). Let next  > 0, and O ⊃ E, O being an open set. There exists F0 ∈ F, such that,  F0 ⊂ O ∩ Ac , and µ(O ∩ Ac ) < ν(F0 ) + . 2 Let F1 ∈ F with F1 ⊂ O ∩ F0c , and  µ(O ∩ F0c ) < ν(F1 ) + . 2 We immediately get F0 ∩ F1 = ∅, and F0 ∪ F1 ⊂ O.

page 22

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Basics of Measure Theory

23

Hence, µ(θ) ≥ ν(F0 ∪ F1 ) = ν(F0 ) + ν(F1 )   ≥ µ(O ∩ Ac ) − + µ(O ∩ F0c ) − . 2 2 Equivalently, µ(O) ≥ µ(O ∩ Ac ) + µ(O ∩ A) −  ≥ µ? (O) ≥ µ? (O ∩ Ac ) + µ(O ∩ A) − . This is true for all open set O ⊃ E. As a result, µ(O) ≥ µ(O ∩ Ac ) + µ(O ∩ A) − . It holds, from all the previous developments, that B ? is a σ-algebra that contains the closed sets in X, and thus it contains the Borel σ-algebra B. Moreover, we observe that µ = µ?|B , and that 1 ≥ µ(X) ≥ sup µ(Kn ) ≥ sup(1 − n

n

1 ) = 1. 2n

Hence, µ(X) = 1, and consequently, µ is a probability measure on X. 2.8

Exercises for Chapter 2

Exercise 1. Let (An )n be a sequence of elements of a σ-algebra F on X. Denote [ \ limAn = ( Ak ), n

n≥0 k≥n

and limAn = n

\ [ ( Ak ). n≥0 k≥n

Show that (1) limAn ⊂ limAn . n

n

(2) limAn ∈ F. n

(3) limAn ∈ F. n

Exercise 2. Let (X, B) be a measurable space, and µ : B → [0 + ∞] be a positive measure on X. Prove the following assertions.

page 23

February 7, 2022 16:4

24

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

(1) Whenever A, B ∈ B such that A ⊆ B, then µ(A) ≤ µ(B). (2) µ is σ-additive,, in the sense that, for any sequence (An )n∈N ⊂ B, we have [ X µ( An ) ≤ µ(An ). n∈N

n∈N

(3) For any increasing sequence (An )n∈N ⊂ B we have µ( lim An ) = lim µ(An ). n→∞

n→∞

(4) Whenever (An )n∈N ⊂ B is a decreasing sequence, for which, there exist an element An0 , such that, µ(An0 ) < ∞, we have µ( lim An ) = lim µ(An ). n→∞

n→∞

(5) For any sequence (An )n≥0 ⊂ B we have µ(lim inf An ) ≤ lim inf µ(An ). n→∞

n→∞

Exercise 3. Prove the assertions 1 to 4 in subsection 2.6. Exercise 4. Let µ be a metric outer measure on (X, d). Let Ann be an increasing sequence of subsets of X with A = limn An , and assume that d(An , A \ An+1 ) > 0, ∀ n. Show that µ(A) = limn µ(An ). Exercise 5. Let Q = (q1 , q2 , ...) be an enumeration of the rational numbers. For A ⊂ R, let X µ(A) = 2−i . qi ∈A

(1) Show that µ is a measure on R. (2) Show that all subsets of R are µ-measurable. (3) Show that µ(R|Q) = 0. Exercise 6. Let µ be a measure on Rn , such that, for all x ∈ Rn , there is a ball B(x, r) with µ(B(x, r)) < ∞. Show that µ is locally finite.

page 24

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Basics of Measure Theory

b4547-main

25

Exercise 7. Let f : [0, 1] → R+ be continuous,, and define µ on [0, 1] by Z µ(A) = f (x)dx. A

Show that µ is equivalent to the Lebesgue measure. Exercise 8. Show that, if µk → µ, and if A is a bounded set with µ(∂A) = 0, where ∂A is the boundary of A, then µk (A) → µ(A). Exercise 9. Let r > 0, and consider in R2 the rectangles of the form Cm,n (r) = (nr, (n + 1)r) × (mr, (m + 1)r), where n, m are integers. Let also µ be a finite measure on R2 , and α ≥ 0. We write Nr (α) = ]{Cm,n (r); µ(Cm,n (r)) ≥ rα }, and define the functions log+ (Nr (α + ) − Nr (α − )) , →0 r→0 − log r

fc (α) ≡ lim lim

f c (α) ≡ lim lim inf

log+ (Nr (α + ) − Nr (α − )) , − log r

f c (α) ≡ lim lim sup

log+ (Nr (α + ) − Nr (α − )) , − log r

→0 r→0

and →0 r→0

for α ≥ 0. Show that fH (α) ≤ f c (α) ≤ f c (α), ∀ α ≥ 0. Exercise 10. A – Let λ be the Lebesgue measure on [−1, 1], and g = χ[0,1] . Let next (fn ) be the sequence of functions defined by  fn (x) = g(x) for n even, fn (x) = g(−x) for n odd. Show that Z 

Z  lim inf fn < lim inf fn . n→∞

n→∞

page 25

February 7, 2022 16:4

26

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

B – Let (X, F, µ) be a measurable space. For a real-valued measurable function f on X, denote Φf (t) = µ({f > t}), t ≥ 0. (1) Show that if f is a step function, then Z Z Φf (t)dt = f dµ. X

R+

(2) Show that for every real-valued measurable function f on X, and for all p > 0, we have Z Z |f |p dµ = p µ({|f | > t})tp−1 dt. X

R+

page 26

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Chapter 3

Martingales with Discrete Time

The notion of martingales constitutes one of the basic concepts applied in fractal analysis. We propose in this chapter to provide a detailed study of such a notion in order to conserve the self-containing aspect of the book. Readers may refer for more details, and for general studies of martingales concept and applications to [Benaych-Georges (2009); Berglund (2014); Berglund and Gentz (2006); Billingsley (1965); Budzinski (2019); Falconer (1985, 1990); Gervais (2009); Rogers (1970)]. 3.1

Generalities

Let (Ω, A, P ) be a probability space. We will denote by L(Ω, A, P ) the vector space of all real A-measurable functions defined on Ω. Such functions are known as random variables. For p ∈ [1, +∞[, we denote Z n o Lp (Ω, A, P ) = f which are A − measurable, and |f |p dP < +∞ . Ω p

We next define the p-norm on L (Ω, A, P ) by Z  p1 kf kp = |f |p dP . Ω

Let B be a complete sub-σ-field in A. As previously, we denote L(B) = {f which are B − measurable}, the vector space of all B-measurable functions on Ω. For 1 ≤ p < ∞, we introduce similarly the Lp (B) vector space by  Lp (B) = f ∈ L(B); kf kp < +∞ . We just recall here that there is no essential difference with the functions that are complex-valued, or taking their values in the higher dimensional 27

page 27

February 7, 2022 16:4

ws-book9x6

28

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Euclidean space Rd , for d ≥ 2. In these cases, the measurability of the whole function is equivalent to the measurability of all its components, and the absolute value will be seen as the module in the complex case or the norm in Rd . As a consequence, all the functions used will be real-valued, except when necessary will recalled. We have the following properties of functional spaces above. Proposition 3.1. (1) L(B) (resp Lp (B), 1 ≤ p ≤ ∞) is the sublinear space of L(Ω, A, P ) (resp Lp (Ω, A, P )) and is reticulated. It is invariant with respect to monotone limits. (2) For 1 ≤ p < ∞, Lp (B) is closed. Proof. Let f, g ∈ L(B). We claim that max(f, g) ∈ L(B) and min(f, g) ∈ L(B). Indeed, for c ∈ R, we have [ (max(f, g))−1 (−∞, c) = f −1 (] − ∞, c[) (g −1 (] − ∞, c[) ∈ B. Similarly, (min(f, g))−1 (−∞, c) = f −1 − ∞, c[)

\

(g −1 (] − ∞, c[) ∈ B.

Hence, L(B) is reticulated. Next, for all f, g ∈ Lp (B), we have k max(f, g)kp = k

f + g + |f − g| kp ≤ kf kp + kgkp < +∞, 2

k min(f, g)kp = k

f + g − |f − g| kp ≤ kf kp + kgkp < +∞. 2

and

Hence, max(f, g) and min(f, g) ∈ Lp (B). Let next (fn )n be Increasing in L(B). We shall show that its limit f ∈ L(B). (Respectively, for (fn )n in Lp (B), we show that its limit f ∈ Lp (B)). We claim that, if M is a sublinear space in Lp (B), satisfying the conditions of Proposition 3.1, there exists a sub-σ-field B of A, such that, M = L(B) (respectively, M = Lp (B)). So, assume, for the second part of the Proposition 3.1, that 1 ≤ p < +∞, so that, Lp (B) will be closed in Lp (Ω, A, P ). Remark 3.1. The space C([0, 1], R) ⊆ L∞ ([0, 1], R), within the Lebesgue measure, is a counter example to the second part of Proposition 3.1, for p = ∞.

page 28

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Martingales with Discrete Time

b4547-main

29

Indeed, the sequence (fn )n defined by  1 , , 0 ≤ x ≤ 21 − 2n 0 1 1 fn (x) = 2nx + 1 − n , ≤ 2 − 2n ≤ x ≤ 12 ,  1 , 12 ≤ x ≤ 1, lies in C([0, 1], R). Its limit being the function  0 , 0 ≤ x < 21 , f (x) = 1 , 12 ≤ x ≤ 1, is not in C([0, 1], R), even though it is in L∞ ([0, 1], R). Corollary 3.1. Let 1 ≤ p < ∞, and U be a positive contractive linear operator on Lp (Ω, A, P ), such that U 1 = 1. Then, there exists a complete sub-σ-field J , satisfying Lp (J ) = {f, U f = f }. Proof. It is straightforward that the set V = {f, U f = f } is a linear subspace of Lp . Moreover, it is closed and contains the constant function 1. Now, as U is a positive operator, we immediately deduce that U (f + ) ≥ U f,

and U (f + ) ≥ 0, ∀ f ∈ Lp .

This yields that U (f + ) ≥ (U f )+ , ∀ f ∈ Lp . Consequently, U (f + ) ≥ f + ≥ 0, ∀ f ∈ V. As a result, due to the fact that U is a contraction, we get U (f + ) = f + , ∀ f ∈ V. Observing now that Sup(f, g) = f + (g − f )+ , it results that V is reticulate. Corollary 3.2. Let {Bn , n ∈ N} be an Increasing sequence of sub-σ-fields S of A, and B∞ = T ( n Bn ). Then, [ Lp (B∞ ) = Lp (Bn ), ∀p, 1 ≤ p < ∞. n∈N

page 29

February 7, 2022 16:4

ws-book9x6

30

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Proof. Observe that

S

Lp (Bn ) is reticulated, closed, and it contains the

n∈N

constant function 1. Hence, by Proposition 3.1, there exists a sub-σ-field B ⊆ A, such that, [ Lp (B) = Lp (Bn ). n∈N

We will show next that B = B∞ . Indeed, {Bn , n ∈ N} being increasing, we S p thus deduce that L (Bn ) is a linear subspace of Lp , which is reticulate, n∈N

and contains the constant function 1. Now, observe that the functions f → af , (f, g) → f +g and (f, g) → f ∨g, (f, g) → f ∧g are continuous. We thus S p L (Bn ) is also a linear subspace of Lp , closed, deduce that the closure n∈N

reticulate, and contains the constant function 1. It results from the previous corollary that there exists a complete sub-σ-field B of A, such that [ Lp (Bn ) = Lp (B). n∈N

Observe next that Lp (Bn ) ⊂ Lp (B), ∀ n ∈ N. As a result, due to the fact that B is complete, we get Bn ⊂ B˜n ⊂ B. Consequently, B∞ ⊂ B, which yields that Lp (B∞ ) ⊂ Lp (B). On the other hand, we have Bn ⊂ B∞ , ∀ n ∈ N. Therefore, Lp (Bn ) = Lp (B∞ )(n ∈ N), which means that Lp (Bn ) ⊂ Lp (B∞ ), (n ∈ N). Hence, Lp (B) =

[

Lp (Bn ) ⊂ Lp (B∞ ).

n∈N

We get finally, Lp (B) = Lp (B∞ ), and that B = B˜∞ .

page 30

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Martingales with Discrete Time

b4547-main

31

Corollary 3.3. Let (Bn )n∈N be a decreasing sequence of sub-σ-fields in A. Then, for 1 ≤ p ≤ ∞, we have \ \ Lp (Bn ) = Lp ( Bn ). n∈N

n∈N

Proof. See Exercise 3.6. 3.2

Conditional expectation

Let B be a sub-σ-field in A, and consider on L2 (Ω, A, P ) the scalar product defined by Z < f, g > = f gdP, ∀f, g ∈ L2 (Ω, A, P ). Ω

We have immediately the main result proved in the theory of Hilbert spaces, stating that for every f ∈ L2 (Ω, A, P ), there exists a unique element f˜ ∈ L2 (B), such that, < f˜, g > = < f, g >, ∀g ∈ L2 (B). f˜ is called the projection of f onto L2 (B), and it satisfies the condition of smallest distance from L2 (B) to f , where the distance between two elements f, g ∈ L2 (Ω, A, P ) is defined by kf − gk2 . In all that follows, we denote f˘ by E B (f ). Definition 3.1. The function E B : f 7−→ E B (f ) is called conditional expectation relatively to B. We immediately have the following results. Proposition 3.2. (1) ∀ f ∈ L2 (Ω, A, P ), ∀ B ∈ B,

Z

E B (f )dP =

B

Z f dP . B

(2) ∀ f ∈ L2 (Ω, A, P ), ∀ h ∈ L∞ (B), E B (hf ) = hE B (f ). Proof. 1. We have for g = XB , Z < f, g >=

f dP B

< E B (f ), g >=

Z

E B (f )dP.

B

2. < hf, g >=< hE B (f ), g >, ∀g.

page 31

February 7, 2022 16:4

ws-book9x6

32

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Z

Z hf gdP =

Ω

f hgdP Ω

B =< Z E (f ), hg > = f E B (f )hgdP Ω

= < hE B (f ), g > . We now prove that hE B (f ) satisfies effectively the properties of characterizing the conditional expectation E B (hf ) of hf . We know that hE B (f ) ∈ L2 (B), and is a product of B – measurable functions is L∞ and L2 , respectively. On the other hand, for any element g ∈ L2 (B), the function hg ∈ L2 (B). As a consequence, Z Z B f E (f ) hg dP = f hgdP, Ω

Ω

which means that E B (f )h − f h is orthogonal to L2 (B). Proposition 3.3. Let U be an orthogonal projector on L2 (B). Then U is a conditional expectation if and only if U is positive, and all constant functions remain invariant within the action of U . Proof. It is easy to observe that E B (1) = 1. Hence, to prove that E B (f ) ≥ 0 for f ≥ 0, we have to use the function 1(E B (f ) 0, P B0 sup Xn ≥ a ≤ min( , 1). a n We write P B (A) to designate E B (1A ), where A is a subset of Ω. ¯ be such that, Proof. Let νa : Ω −→ N   min{n, Xn (w) > a} if (sup Xn )(w) > a, n νa (w) =  ∞ if (sup Xn )(w) ≤ a. n

It holds, ∀p ∈ N, that n o n o νa = p = w, Xn (w) > a, ∀ n ≥ p, Xm (w) ≤ a, ∀ m ≤ p − 1  \ \  \ {Xn ≤ a} ∈ Bp . {Xn > a} = m≤p−1

n≥p

For p = ∞, we have n o n o \n o νa = ∞ = w/Xn (w) < a, ∀ n = Xn ≤ a ∈ B∞ . n

Then, νa is a stopping time. As on {νa < +∞}, we have Xνa > a, and the constant a defines a positive upper martingale, it holds that (Yn )n given by Yn = Xn for n < ν

and

Yn = a for n ≥ νa ,

is a positive upper martingale. We thus have Y0 ≥ E B0 (Yn ) ⇒ X0 ≥ E B0 (Yn )

a > E B0 (Yn ).

and

Consequently, min(X0 , a) ≥ E B0 (Yn ) ≥ E B0 (a1νa ≤n ). As a result, min(X0 , a) ≥ aE B0 (1νa≤n ) = aP B0 (νa ≤ n). Letting n −→ ∞, we obtain min(X0 , a) ≥ aP B0 (νa < ∞) = aP B0 (sup Xn > a). n

Hence, min(

X0 , 1) ≥ P B0 (sup Xn > a). a n

page 42

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Martingales with Discrete Time

43

Corollary 3.6. As a consequence of the previous results, we have the following. Z X0 , 1)dP . (1) P (X0 < ∞, sup Xn > a) ≤ min( a n {X0 a) ≤ , 1)dP. min( a n {X0 a, ∀n), we put νk = ∞ as well as all νj , j ≥ k. We designate by βa,b the greatest integer p, such that, ν2p is finite . βa,b = ∞ whenever all νk are finite . βa,b designates in fact the number of times that the sequence (xn )n crosses the interval [a, b]. We have the following result. Proposition 3.9. (1) limxn < a < b < limβa,b ⇒ βa,b = ∞ ⇒ limxn ≤ a < b ≤ limxn . n

n

n

n

(2) The sequence (xn )n is convergent ⇐⇒ βa,b < ∞, ∀a, b ∈ R.

page 43

February 7, 2022 16:4

44

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Consider now a sequence (Xn )n of positive continuous random variables. It holds that the positive real numbers a, b suffice instead of taking them with arbitrary signs in R. By the previous procedure, for any element w ∈ Ω, we associated a sequence of indices (νk (w))k , which are associated in turn to the real sequence (xn = Xn (w))n , and thus, we defined a sequence of random variables (νk )k . We obtain the following characterizations for the sequence (νk )k , [ {ν2p = n} = {ν2p−1 = n, Xm+1 < b, ..., Xn−1 < b, Xn ≥ b}, m 0, denote E = {x ∈ Rd ; kx − yk ≤  f or some y ∈ E}, and similarly, F = {x ∈ Rd ; kx − yk ≤  f or some y ∈ F }. The Hausdorff distance d(E, F ) is evaluated as d(E, F ) = inf{ > 0; E ⊆ F and F ⊆ E }. We will see later that this distance makes the set C(Rd ) of all compact sets in Rd a complete metric space. Proposition 4.1. For α > 0 fixed, i. Hα is an outer metric measure on Rd . ii. Hα is regular. Proof. i. We have to show that a. Hα (∅) = 0. b. Hα is monotone, in the sense that Hα (E) ≤ Hα (F ), whenever E ⊆ F ⊆ Rn . c. Hα is sub-additive, in the sense that X Hα ( ∪ Ep ) ≤ Hα (Ep ). p≥0

p≥0

for all Countable family (Ep )p≥0 of subsets of Rn . Assertion a. is obvious. We shall prove b. Let E ⊆ F be subsets of Rn . It is straightforward that any -covering (Uj ) of F is obviously an -covering of E. Consequently, X Hα (E) ≤ |Uj |α . j

Taking the inf on all the -covering (Uj ) of F we obtain Hα (E) ≤ Hα (F ),

page 51

February 7, 2022 16:4

ws-book9x6

52

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

which yields, with the limit on , that Hα (E) ≤ Hα (F ). We now prove assertion c. Let  > 0, (Ep )p be a Countable family of subsets Ep ⊂ Rd . Without loss of the generality, we may assume that X Hα (Ep ) < ∞. p≥0

For p ∈ N fixed, let (Up,j )j be an -covering of Ep , such that, X  |Up,j |α ≤ Hα (Ep ) + p . 2 j S It is obvious that (Up,j )j,p is an -covering of E = p Ep , and thus X[ XX |Up,j |α ≤ Hα (E) ≤ Epα (Ep ) + , p

p

j

p

which, by letting  → 0, yields assertion c. We now prove that Hα is metric. Let E, and F be subsets of Rd , such that d(E, F ) > 0, where d is the Hausdorff distance of sets. Let also (Uj )j∈N d(E, F ) . We immediately observe be an -covering of E ∪ F , with 0 <  < 2 that, for all j ∈ N, Uj ∩ E = ∅

or Uj ∩ F = ∅.

Therefore, for α ≥ 0, we get X X |Uj |α = j

|Uj |α +

j; Uj ∩E=∅

X

|Uj |α .

j; Uj ∩F =∅

Next, as X

|Uj |α ≥ Hα (E),

j; Uj ∩E=∅

and similarly, X

|Uj |α ≥ Hα (F ),

j; Uj ∩F =∅

we obtain X

|Uj |α ≥ Hα (E) + Hα (F ),

j

for all -covering (Uj )j∈N of E ∪ F . Consequently, Hα (E ∪ F ) ≥ Hα (E) + Hα (F ), ∀ ,

page 52

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Hausdorff Measure and Dimension

b4547-main

53

which, by letting  ↓ 0, yields that Hα (E ∪ F ) ≥ Hα (E) + Hα (F ). The opposite inequality is always true due to the sub-additivity property of Hα , and does not require the assumption d(E, F ) > 0. As a result, we get Hα (E ∪ F ) = Hα (E) + Hα (F ) whenever d(E, F ) > 0. It remains to prove that Hα is regular. So, let E ⊂ Rd . We shall prove that there exists a Hα -measurable set A ⊂ Rd , such that, Hα (E) = Hα (A). Observe that whenever Hα (E) = ∞, the problem is solved by choosing A = Rd . So, the interesting case is when Hα (E) < ∞. Assume that this 2 situation occurs, fix n ∈ N, and let n = . There exists an n -covering n (Vn,j )j of E composed of open sets, and satisfying X 1 |Vn,j |α ≤ H α1 (E) + . n n j Consider next A=

\[ n

Vn,j .

j

It is straightforward that A is a Borel set. It is therefore Hα -measurable. Moreover, E ⊂ A. Hence, Hα (E) ≤ Hα (A). Besides, A⊂

[

Vn,j , ∀n.

j

Hence, (Vn,j )j is an n -covering of A. Consequently, X 1 Hαn (A) ≤ |Vn,j |α ≤ H α1 (E) + , ∀n. n n j As a result, Hαn (A) ≤ H α1 (E) + n

1 , ∀n. n

Letting n −→ ∞, this leads to Hα (A) ≤ Hα (E). So, finally, we get Hα (A) = Hα (E).

page 53

February 7, 2022 16:4

ws-book9x6

54

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Definition 4.1. The restriction of Hα on the σ-algebra of Hα -measurable sets is called the Hausdorff measure with dimension α, or also the αHausdorff measure. Proposition 4.2. The following assertions are true. (1) For E ⊂ Rd , and 0 < α < β, we have Hα (E) < ∞ =⇒ H β (E) = 0. (2) Whenever α > d, we get Hα (Rd ) = 0. Proof 1. For any -covering (Uj )j of E, we get X X |Uj |β ≤ β−α |Uj |α . Hβ (E) ≤ i

i

This leads to Hβ (E) ≤ β−α Hα (E). Letting  ↓ 0, we obtain H β (E) = 0. 2. We will prove that H β (C) = 0, for any cube C of Rd , with unit side. Indeed, write d

C=

k [

Ck,j ,

j=1

√ d 1 , where, Ck,j are cubes with side length , k ∈ N. Next, denote k = k k and observe that √ !α X d α α d Hk (C) ≥ , ∀ k. |Ck,j | = k k j Hence, letting k −→ ∞, we get Hα (C) = 0.

Corollary 4.1. For all E ⊂ Rd , there exists a unique critical value αE ≥ 0, satisfying • Hα (E) = 0, ∀ α > αE . • Hα (E) = ∞, ∀ α < αE .

page 54

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Hausdorff Measure and Dimension

55

Proof. Denote αE = inf{α > 0; Hα (E) = 0}. As the last set is not empty (it contains d + 1), and is contained in (0, ∞), so, its lower bound exists in [0, ∞]. Denote it by αE . We get immediately assertions 1, and 2. Definition 4.2. The critical value αE is called the Hausdorff dimension of E, and denote dimH E or simply dimE. Figure 4.1 illustrates the concept of the Hausdorff dimension graphically.

Fig. 4.1: The Hausdorff dimension of a set E.

Proposition 4.3. The following properties hold. (1) (2) (3) (4)

0 ≤ dim E ≤ d, ∀ E ⊂ Rd . dim E ≤ dim F , ∀ E ⊂ F ⊂ Rd . dim{x} [ = 0, ∀ x ∈ R. dim En = sup dim En , ∀ (En )n a sequence of subsets of Rd . n

n

(5) dim E = 0 whenever E is a finite or Countable set in Rd . (6) ∀ E ⊂ Rd , 0 < Hα (E) < ∞ =⇒ dim E = α. (7) ∀ E ⊂ Rd , such that λ(E) > 0, we have dim E = d, where λ is the Lebesgue measure on Rd . (8) ∀ E ⊂ Rd , 0 ≤ H dimE (E) ≤ ∞.

page 55

February 7, 2022 16:4

ws-book9x6

56

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Proof. 1. dimE ≥ 0 follows from its definition as a lower bound of a set of positive real numbers. On the other hand, as E ⊂ Rd , and Hα (Rd ) = 0 for all α > d (See Proposition 4.2), we get Hα (E) = 0 for all α > d. Hence, dim E ≤ d. 2. Assume that dimF < dimE, and let η > 0 small enough be such that, α = dimF + η < dimE. α

It follows that H (E) = ∞. Consequently, as E ⊂ F , we get Hα (F ) = ∞, which contradicts the definition of dimF . Hence, the assertion follows. 3. ∀ x ∈ Rd , and ∀ > 0, the set (B(x, )) composed of the single Ball is an -covering of {x}. Henceforth, Hα ({x}) ≤ α , ∀α > 0. Thus, Hα ({x}) = 0, ∀α > 0. Consequently, dim{x} = 0. 4. Recall that from the sub-additivity property of the Hausdorff measure, we may write, for all α > 0, [ X H α ( En ) ≤ Hα (En ). n

n

Consequently, [ Hα ( En ) = 0, ∀α > sup dimEn . n

n

Which leads immediately to [ dim( En ) ≤ sup dimEn . n

n

On the other hand, whenever α < supn dimEn , there exists at least one set En0 , for some n0 , such that, α < dimEn0 . For this [ particular set, we have α immediately H (En0 ) = ∞. So next, as En0 ⊂ En , we get n

[ H ( En ) = ∞. α

n

This yields that [ α < dim( En ), ∀α < sup dimEn . n

n

Consequently, [ sup dimEn ≤ dim( En ). n

n

page 56

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Hausdorff Measure and Dimension

b4547-main

57

5. This follows from assertions 3, and 4. 6. Assume that this does not occur. So, α < dimE or α > dimE. In the first case, we get Hα (E) = ∞, and in the second, we get Hα (E) = 0. As both consequences are not true, we get necessarily α = dimE. 7. For all  > 0, and all -covering (Uj )j of E, we get X X λ(E) ≤ λ(Uj ) ≤ |Uj |d . j

j

Consequently, λ(E) ≤ Hd (E), ∀ > 0. As a result, λ(E) ≤ H d (E). Now, as λ(E) > 0 we get H d (E) > 0. Which means that dimE ≥ d. Finally, observing assertion 1, we get dimE = d. 8. To prove the assertion, we shall construct sets E for which 0, and ∞ may be reached. It suffices to take E = Rd , and E = Qd . 4.2

Hausdorff dimension of Cantor-type sets

The main problem in the theory of Hausdorff measure, and dimension is the computation of the such dimension using its original definition. In the present section, we propose to evaluate the Hausdorff dimension of some particular sets known as the Cantor’s type. For the sake of simplicity, we will focus on the triadic well known Cantor’s set. The readers may adopt the method developed for general Cantor’s sets. The construction of the triadic Cantor set starts from the unit interval I0 = [0, 1]. Next, in a first step we split I0 into three intervals, 1 1 2 2 I00 = [0, ], I01 = [ , ], and I02 = [ , 1]. 3 3 3 3 We keep next the first, the third ones, and delete the middle one. We get so, two intervals 1 2 I00 = [0, ], and I02 = [ , 1]. 3 3 Next, at the second step we split each one of the last intervals into three sub-intervals always with the same length. We get for I00 , 1 1 2 2 3 I000 = [0, ], I001 = [ , ], and I002 = [ , ], 9 9 9 9 9

page 57

February 7, 2022 16:4

58

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

and similarly for I02 , 7 8 8 6 7 I020 = [ , ], I021 = [ , ], and I022 = [ , 1]. 9 9 9 9 9 Keep next the first, the third ones, and delete the middle for both subdivisions of I00 , and I02 . We get so, 4 intervals, 1 2 3 6 7 8 I000 = [0, ], I002 = [ , ], I020 = [ , ], and I022 = [ , 1]. 9 9 9 9 9 9 We continue the process similarly. So, given an integer n ∈ N, we get at the 1 step n a number 2n of intervals with the same length n of the form 3 ak ak + 1 In,k = [ n , ], 3 3n for some integer ak . Figure 4.2 below illustrates the process.

Fig. 4.2: The triadic Cantor set.

Next, denote for n ∈ N, n

En =

2 [

In,k .

k=1

Definition 4.3. The triadic Cantor set is defined by C =

\ n≥0

En .

page 58

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Hausdorff Measure and Dimension

59

The following are special characteristics of the triadic Cantor set C, and give a first example of exact calculus in fractal geometry. Lemma 4.1. For all k ∈ N, the set Ek is the union of 2k intervals Iak = [

ak ak + 1 , ], 3k 3k

where ak is of the form ak =

k X

xi 3k−i , xi ∈ {0, 2}, ∀ i.

i=0

Proof. We proceed by recurrence on the integer k. Indeed, for k = 0, E0 = [0, 1], let a0 = 0, and Ia0 = E0 . Assume next that Ek is a union of 2k intervals ak ak + 1 ]. Iak = [ k , 3 3k The next step in the construction of the triadic Cantor set consists of subdividing each interval of Ek into 3 sub- intervals with the same length, and omitting the middle one. This means that Iak is transformed into a union Ia1k ∪ Ia2k , where Ia1k = [

ak ak + 1 , ], 3k+1 3k+1

and ak + 2.3k ak + 2.3k + 1 , ]. 3k+1 3k+1

Ia2k = [

We thus get Ek+1 as a union of 2k+1 intervals of the form above with ak+1 = ak , or ak+1 = ak + 2.3k , which guaranties that for ak+1 also the xi ’s are 0 or 2. Corollary 4.2. C=

X ∞ i=1

 ai ; ai ∈ {0, 2}, ∀ i . 3i

Indeed, this follows from the last Lemma. Let x ∈ C. So, x ∈ Ek for all k. Consequently, there exists a sequence ak =

k X i=0

xi 3k−i , xi ∈ {0, 1, 2}, ∀ i.

page 59

February 7, 2022 16:4

60

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

such that, x ∈ Iak . Which yields otherwise that k

k

X xi ak + 1 X xi 1 ak = ≤x≤ = + k. k i k i 3 3 3 3 3 i=0 i=0 Letting k → ∞ we get x=

+∞ X xi i=0

3i

.

The following proposition resumes some topological properties of the triadic Cantor set C. Proposition 4.4. The triadic Cantor set C is non-empty, compact, perfect, with empty interior, non Countable, and with Lebesgue measure zero. Proof. i. C is non empty as it contains 0, and 1. T ii. C is closed as C = Ek , which is an intersection of closed intervals. It k≥0

is obviously bounded as C ⊂ [0, 1]. So; it is compact. iii. The interior of C is empty. We proceed by the converse. So, whenever C contains an interval of length  > 0, such interval will be surely contained in Ek , ∀ k, and consequently contained in one of the intervals Iak , ∀ k. As a result, 1 0 <  < k , ∀ k, 3 which is contradictory. iv. C is perfect. Here also we use the converse reasoning. Assume that C is not perfect. Hence, there exists real numbers a < b such that [a, b] ⊂ C. So, [a, b] ⊂ Ek , ∀ k ≥ 0. Thus, there exists ak such that [a, b] ⊂ Iak . This yields that 1 b − a ≤ ( )k ∀ k, 3 which is impossible. v. C is non Countable. Indeed, if it is Countable, we may write it in a Countable way C = {x1 , x2 , . . . , xn , . . . }. We will construct another element of C which is different from all these xk ’s. To do it, we consider the triadic representation of the elements x1 , . . . , xn , . . . . Let next x = a1 , a2 , . . . , an , . . . be the triadic real number constructed as follows. The element a1 ∈ {0, 2} is chosen to be different from the first digit after the

page 60

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Hausdorff Measure and Dimension

61

decimal point of x1 . It is always possible as the digits xi,1 of x1 are 0 or 2. If x1,1 = 0, we take a1 = 2, and for x1,1 = 2, we take a1 = 0. Next, we chose the digit a2 ∈ {0, 2} to be different from the second digit after the decimal point of x2 , and so on. We obtain, consequently, an element x ∈ C different from all the elements xk ’s, which is contradictory vi. Denote m the Lebesgue measure on R. We have m(C) ≤ m(Ek ) ≤

X

(Iak ) =

m

k 2X −1

1 2 ( )k = ( )k −→ 0, 3 3 i=0

as

k → ∞.

Theorem 4.1. dim C =

log 2 . log 3

log 2 . In one hand, we have C ⊂ En , ∀ n. Conselog 3 1 quently, (In,k )1≤k≤2n is an n -covering of C, where n = n . Hence, 3  α 2n X 1 α Hn (C) ≤ |In,k |α = 2n = 1. 3n

Proof. Denote α =

k=1

As a result, Hα (C) ≤ 1. As a consequence, α ≥ dim C. We now prove the opposite inequality. Let γ,  ∈ R be such that 1 and γ < 1. 3 Let also (Uj )j be an -covering of C, and consider next a sequence (Lj )j of open intervals, such that, γ > 1, 0 < 
0, ∃ -covering of E by elements from F.

(4.1)

n

For E ⊂ R , and α,  > 0, denote Hα (E, F) = inf

X

|Uj |α ,

j

where the lower bound is taken over all the -coverings of E by elements (Uj )j ⊂ F. As for the case of the Hausdorff measure, we get here a Decreasing function relatively to . So, denote Hα (E, F) = lim Hα (E, F). →0

It consists here-also of an outer, and regular metric measure Hα (., F) on Rd . It yields a dimension for sets as n o dimF E = inf α > 0; Hα (E, F) = 0 . Definition 4.4. The collection F is said to permit the computation of the Hausdorff dimension of sets in Rd iff dimF E = dimE, ∀E ⊂ Rn . We will expose now an example of collections F of subsets of Rn permitting the computation of the Hausdorff dimension. Example 4.1. Consider on [0, 1[ a collection F of sub-intervals, satisfying S a. F = n≥0 Fn . b. ∀n, Fn is a finite partition of [0, 1[ on semi-open intervals closed at the left. c. Fn+1 is a refinement of Fn , in the sense that, any element I ∈ Fn+1 is strictly contained in one element p(I) ∈ Fn , called its father or predecessor.

page 63

February 7, 2022 16:4

ws-book9x6

64

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

d. ∀x ∈ [0, 1[, ∀ > 0 there exists I ∈ F such that x ∈ I, and |I| ≤ . e. ∀α > 0, we have n o lim sup |I|α K(I), I ∈ F, |I| ≤  ≤ 1, →0

where o n |I| , J ∈ F, p(J) = I . K(I) = sup |J| Hence, the collection F permits the computation of the Hausdorff dimension on [0, 1[. Otherwise, dimF E = dim E, ∀E ⊂ [0, 1[. Indeed, let I ⊂ [0, 1[ be such that, n o |I| < inf |J|; J ∈ F0 . One of the following assertions holds. i. There exists J ∈ F such that J ⊂ I ⊂ p(J). ii. There exists J1 , J2 ∈ F disjoint such that [ [ J1 J2 ⊂ I ⊂ p(J1 ) p(J2 ). Indeed, let J1 ∈ F such that J1 ⊂ I. Whenever p(J1 ) + I, and J1 has a minimal order k, and thus J1 ∈ Fk consider H ∈ Fk−1 contiguous to p(J1 ). There exists J2 contiguous to p(J1 ), such that, J2 ⊂ I, and p(J2 ) * I. We immediately get [ [ J1 J2 ⊂ I ⊂ p(J1 ) p(J2 ). So, assertion ii holds. Next, observe that ∀ > 0, ∃η > 0, such that, ∀I ∈ F, |I| < η ⇒ p(I) < .

(4.2)

Denote an = inf{|I|, I ∈ Fn }. It is straightforward that an & 0 as n → ∞. On the other hand, an ≥ l > 0 ∀n. Denote En =

[

{I ∈ Fn ; |I| ≥ l}.

T It consists of a decreasing sequence of sets. Let next E = En . As m(En ) ≥ l, it holds immediately that m(E) > 0. Therefore, there exists t ∈ [0, 1[,

page 64

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Hausdorff Measure and Dimension

b4547-main

65

such that, |In (t)| > 0, which is contradictory. Otherwise, ∀α > 0, there exists Mα , such that, |I|α K(I) < Mα , ∀I ∈ F. Now, for  > 0, and η defined in (4.2), denote E = intervals contained in [0, 1[. Denote also

S

j Ij ,

where Ij are

L1 = {Ij ; Ij satisfies assertion i.}, and L2 = {Ij ; Ij satisfies assertion ii.}. Whenever I ∈ L1 , there exists J ∈ F, such that, J ⊂ I ⊂ p(J). Similarly, whenever I ∈ L2 , there exists J1 , J2 disjoint in F, satisfying [ [ J1 J2 ⊂ I ⊂ p(J1 ) p(J2 ), and [ [ [ E ⊂ ( p(J)) (∪p(J1 ) (∪p(J2 )). Now observe that I ∈ L1 , and |J|, |J1 |, |J2 | < η, for η > 0 small enough. Consequently, |p(J)|, |p(J1 )|, |p(J2 )| < . We thus get an -covering of E by elements of F. Moreover, for all β > 0, we have X X |p(J)|α(1+β) = |p(J)|α |p(J)|αβ I∈L1 I∈L1  α X β |p(J)| |J|α = |p(J)| |J| I∈L1 X = Mβα |I|α . I∈L1

Similarly, we get X

|p(J1 )|α(1+β) ≤ Mβα

I∈L2

X

|I|α ,

I∈L2

and X

|p(J2 )|α(1+β) ≤ Mβα

I∈L2

X

|I|α .

I∈L2

As a result of these estimations, we obtain X X |p(J)|α(1+β) ≤ 3 Mβα |I|α . I∈L1 ∪L2

I

page 65

February 7, 2022 16:4

ws-book9x6

66

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

This yields that Hα(1+β) (E, F) ≤ 3Mβα Hηα (E). Letting  ↓0 , this implies that Hα(1+β) (E, F) = 0. Therefore, dimF E ≤ α(1 + β), ∀β > 0. Consequently, dimF E ≤ α. 4.4

Upper and lower bounds of the Hausdorff dimension

In this part, we propose to present an upper bound for the Hausdorff dimension of sets. Let E be a bounded subset in Rd . For  > 0, denote N (E) the minimum number of Balls with diameter  that cover E. Denote also δ(E) = lim inf &0

log N (E) . − log 

The following result shows an upper bound of the Hausdorff dimension of sets. Proposition 4.5. For all bounded set E ⊂ Rd , we have dimE ≤ δ(E). Proof. Let η > 0, and α > δ(E). Consider also a collection of Balls (Bj )1≤j≤N (E) , such that, |Bj | =  ≤ η. It holds that X Hηα (E) ≤ |Bj |α = N (E)α . ∀, 0 <  < η. j

Next, as δ(E) < α, we get N (E)α ≤ 1, for some η > 0 small enough, and for all , 0 <  < η. Consequently, Hηα (E) ≤ 1, which yields that Hα (E) < ∞. Hence, dimE ≤ α, ∀ > δ(E). Thus, dimE ≤ δ(E).

page 66

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Hausdorff Measure and Dimension

67

Now, similarly to the previous case, we aim to give a lower bound for the Hausdorff dimension of sets. To do this we introduce some useful concepts that will be used. Definition 4.5. A Borel measure µ on Rd is said to be α-H¨older (α ≥ 0) iff there exists a constant C > 0, such that, µ(B) ≤ C|B|α , for any Ball B in Rd . We say also that µ is α-H¨olderian, or is H¨olderian with exponent or index α. The following proposition shows a lower bound for the Hausdorff dimension of sets in some special cases. Proposition 4.6. Let E be a Borel subset in Rd , for which, there exists a H¨ olderian measure µ of index α on Rd , such that, µ∗ (E) > 0. Then, α H (E) > 0, and consequently, dimE ≥ α, where nX o [ µ∗ (E) = inf µ(Uj ); Uj is bounded ∀j, and Uj ⊃ E . j

j

Proof. Consider for  > 0 an -covering of E by means of Balls (Bj )j . It holds that X 1 X 1 ∗ µ (E) ≤ µ(Bj ) ≤ |Bj |α . C C j j This yields that 0
0, which implies that dimE ≥ α. Proposition 4.7. Let E ⊂ Rd , α ≥ 0, and Φ : Rd → Rd be such that, |Φ(x) − Φ(y)| ≤ C|x − y|α , ∀x, y ∈ Rd . Then, dimΦ(E) ≤

1 dimE. α

The function Φ above is said to be α-H¨ older or α-H¨ olderian or H¨ olderian with exponent or index α.

page 67

February 7, 2022 16:4

68

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Proof. Denote γ = dimE, and for η > 0, denote γ s = (1 + η). α Consider next, for  > 0, an -covering (Ij )j of E, and denote δ = Cα . It holds immediately that |Φ(Ij )| ≤ C|Ij |α . This yields that (Φ(Ij ))j is a δ-covering of Φ(E). As a result Hδs (Φ(E)) ≤ C s Hγ(1+η) (E). Letting  ↓ 0, we obtain Hs (Φ(E)) = 0, which means that s ≥ dimΦ(E). Letting now η ↓ 0, we obtain γ ≥ dimΦ(E). α

4.5

Frostman’s Lemma

It is a basic result in measure theory, which is applied widely in multifractal analysis. It permits to construct measures of Frostman’s type on multifractal sets whenever Gibbs hypothesis is no longer valid. Theorem 4.2. Let A ⊂ Rd be a compact set, α ≥ 0, such that, Hα (A) > 0. Then, there exists a Borel probability measure µ on Rd , supported on A, and satisfying µ(B) ≤ M |B|α , for all Ball B in Rd . Hα being the α-Hausdorff measure. Proof. We will split the proof into steps. Claim 1. There exists a constant γ > 0 such that X |Uj |α ≥ γ, j

for all covering (Uj )j of A. Claim 2. There exists a sequence (µn )n of Borel probability measures on Rd , such that, µn (C) ≤ M |C|α ,

page 68

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Hausdorff Measure and Dimension

69

for all dyadic cube C of order ≤ n, and µn (En ) = 1, where En is the union of all dyadic cubes of order n intersecting A. Claim 3. The sequence (µn )n is relatively compact, so that, it has a subsequence (µnk )k converging weakly to a measure µ. Proof of claim 1. Observe that Hα (A) = lim Hα (A). ↓0

As a result, there exists  > 0, such that, Hα (A) > 0. Let (Uj )j be a covering of A. We have obviously sup |Uj | ≤  or sup |Uj | > . j

j

Consequently, X

|Uj |α ≥ min{α , Hα (A)} = γ.

j

Proof of claim 2. Let n ∈ N, be fixed, and consider the dyadic cubes of order n intersecting A. Consider next the measure µnn defined by 1 α ) if C ∩ A 6= ∅, 2n and 0 iff C ∩ A = ∅. Next, at the order n − 1, the weights will be evaluated according to µnn . For a dyadic cube C of order (n − 1) intersecting A, we put µnn (C) = (

µn−1 (C) = µnn (C) whenever µnn (C) ≤ ( n When the same cube is such that µnn (C) > (

1 2n−1

)α .

1 α ) , we put 2n−1

µnn−1 (C) = λ µnn (C), where 0 < λ < 1 is such that, µn−1 (C) = ( n

1 α ) . 2n−1

Of course, when C ∩ A = ∅, we put µn−1 (C) = 0. We thus get n  1  µn−1 (Cn ) ≤ ( n )α if C is a dyadic cube of order n. n 2 1  µn−1 (C) ≤ ( n−1 )α if C is a dyadic cube of order n − 1. n 2

page 69

February 7, 2022 16:4

ws-book9x6

70

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

We follow the process until we get µ1n . This latter is a finite measure on Rd , and moreover, it satisfies 1 µ1n (C) ≤ ( k )α , ∀ C a dyadic cube of order k ≤ n, 2 and µ1n (En ) = ||µ1n || < ∞, where En is the union of all dyadic cubes of order n intersecting A. We put next 1 µn = µ1 . ||µ1n || n Observe next that ∀t ∈ A, there exists a dyadic cube C of order k ≤ n, such that, t ∈ C, and 1 µ1n (C) = ( k )α , 2 which may be written as √ !α 1 α d 1 . µn (C) = ( √ ) k 2 d √ 1 α d Denoting β = ( √ ) , and observing that |C| = k , we get 2 d µ1n (C) = β|C|α . Next, for t ∈ A, let Ct be the largest dyadic cube of order ≤ n containing t, and satisfying µn (Ct ) = β|Ct |α . We immediately observe that whenever t0 6= t in A, we get either Ct = Ct0 or Ct ∩ Ct0 = ∅. Hence, as A is compact, and A ⊂ ∪t∈A Ct , we may cover it with a finite number p of cubes Ctj , 1 ≤ j ≤ p (the cubes being dyadic, with orders ≤ n, and disjoint). Consequently, [ X X ||µ1n || ≥ µ1n ( Ctj ) = µ1n (Ctj ) = β |Ctj |α ≥ βγ. j

j

j

As a result, µn (C) ≤

1 1 β|C|α = |C|α βγ γ

page 70

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Hausdorff Measure and Dimension

b4547-main

71

for all dyadic cube C of order ≤ n. Furthermore, µn (En ) = 1. Denote next, Γ = {µn ; n ∈ N}. It is straightforward that En ⊂ En−1 ⊂ · · · ⊂ E1 . Consequently, µn (E1 ) = 1, ∀n. Hence, Γ is tight. Consequently, it is relatively compact. Hence, there exists a sub-sequence (µnk )k of (µn )n converging in the weak sense to a Borel measure µ on Rd . We will prove now that µ is α-H¨olderian. Let d Y mj mj + 1 C= [, mj ∈ Z, [ n, 2 2n j=1 be a dyadic cube of order n, and let d Y mj − 1 nj + 2 e= [ C , [. 2n 2n j=1 Let next f : Rd −→ [0, 1] be continuous, such that f ≡ 1 on C, support(f ) ⊂ e and 0 ≤ f ≤ 1. We have C, Z Z e µ(C) ≤ f dµ = lim f dµnk ≤ lim µnk (C). k→+∞

C

k→+∞

e C

e is a union of 3d dyadic cubes of order n of type C, we get As C 1 µ(C) ≤ 3d |C|α . γ Let now B(x, r) be a Ball in Rd . If 2r < 1, let n be such that, 1 1 ≤ 2r ≤ n−1 . 2n 2 B(x, r) is contained in at most 3d dyadic cubes Cj of order n. Consequently, 1 9d √ µ(B(x, r)) ≤ 3d µ(Cj ) ≤ 3d 3d |Cj |α = ( d)α |2r|α . γ γ 9d √ α Denoting M = ( d) , this means that γ µ(B(x, r)) ≤ M |2r|α . It remains finally to show that µ(A) = 1. Indeed, as A = ∩∞ n=1 En , we get µ(A) =

lim

n−→+∞

µ(En ).

On the other hand, we have µ(En ) ≥ lim sup µnk (En ) = 1. k→+∞

Consequently, µ(A) = 1.

page 71

February 7, 2022 16:4

72

4.6

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Application

Q Let {bj } ⊂ N, bj ≥ 2, and denote aj = 1≤k≤j bk . For all x ∈ [0, 1], consider its expression as X xj with xj ∈ {0, 1, 2, . . . , bj−1 }. x= aj j≥1

Consider next the sets n o X 2j+1 E= x= , 0 ≤ 2j+1 < b2j+1 , a2j+1 j≥0

and o n X 2j , 0 ≤ 2j < b2j . F = x= a2j j≥0

Then E, and F are compact. We now show that for a suitable choice of (bj ), we may obtain dimE = dimF = 0. For n ∈ N, consider the collection of intervals in [0, 1[ defined by n n n hX xj X xj 1 io F n = In = , + . a a an j=1 j j=1 j It is easy to see that {Fn }n is a partition of [0, 1[, and that for all n, Fn+1 is a refinement of Fn . Moreover, for 0 ≤ x < 1, there exists n, such that, x ∈ In . Denote as previously nX  o j E= , 0 ≤ j < bj+2p = 0 , aj j and F =

o nX  j , 0 ≤ j < bj+2p+1 = 0 . aj j

We obtain here compact subsets E, and F in [0, 1[. Let next A = { = (n ), n ∈ {0, 1, 2, . . . , bn−1 }}, and d be the distance defined on A as follows, ? d(, 0 ) = 0 whenever  = 0 . ? d(, 0 ) = 1 whenever 1 6= 01 (and thus  6= 0 ). 1 ? d(, 0 ) = whenever 1 = 01 ..., n = 0n , and n+1 6= 0n+1 . an

page 72

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Hausdorff Measure and Dimension

73

The map d constructed above is an ultra-metric distance, as it satisfies particularly d(, 0 ) ≤ sup{d(, ”), d(”, 0 )}. We may show that i. (A, d) is compact. ii. The sets e = { = (j )/2p = 0}, E and Fe = { = (j )/2p+1 = 0} are closed in A, and thus compact. Consider the function ϕ : A → [0, 1] X j .  7→ aj j≥1

We immediately notice that e = E, ϕ(E) and ϕ(Fe) = F. Consequently, it suffices to show that ϕ is continuous, particularly, |ϕ() − ϕ(0 )| ≤ d(, 0 ). For  = 0 , the inequality is satisfied. For 1 6= 01 , the inequality is also satisfied. Whenever 1 = 01 , . . . , n = 0n , and n+1 6= 0n+1 , we have |ϕ() − ϕ(0 )| ≤

1 = d(, 0 ). an

We now recall that dimE ≤ δ(E) = lim inf →0

log N (E) . − log 

On the other hand, N a1 (E) ≤ 2k

k Y j=1

b2j−1 .

page 73

February 7, 2022 16:4

ws-book9x6

74

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Consequently, dimE ≤ δ(E) ≤ lim inf

log N a1 (E) 2k

log a2k

k→+∞

.

Henceforth, k X

dimE ≤ limk

log b2j−1

j=1 2k X

. log bj

j=1 j!

Taking bj = 22 , we obtain dimE = dimF = 0. 4.7

Exercises for Chapter 4

Exercise 1. Consider the collection F composed Balls B(x, r) in Rd . Show that F permits the computation of the Hausdorff dimension of sets in Rd . Hint. We may prove firstly that, for all α ≥ 0, we have Hα (E, F) ≤ 2α Hα (E) ≤ 2α Hα (E, F). Exercise 2. Show that the collection F of dyadic intervals in R permits the computation of the Hausdorff dimension. Recall that, for an integer c ≥ 2, we call c-c-adic interval of order n ∈ N any interval of the form p p+1 Ipn (c) = [ n , n [, p ∈ Z. c c Hint. We may prove as in Exercise 1 that, for all α ≥ 0, we have Hα (E, F) ≤ 3Hα (E) ≤ 3Hα (E, F). Exercise 3. Let F ⊂ Rd , and Φ : F → Rd be such that, C2 |x − y| ≤ |Φ(x) − Φ(y)| ≤ C1 |x − y|, ∀x, y ∈ F, (C1 , C2 > 0 constants). Show that dimΦ(F ) = dimF . Exercise 4. Let F ⊂ Rd be such that, dimF < 1. Show that F is totally discontinuous: Each connected component of F is reduced to one point.

page 74

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Hausdorff Measure and Dimension

b4547-main

75

Hint. We may consider, for x ∈ F fixed, the function f : Rd → [0, +∞[, such that, f (z) = |z −x|, and prove next, for x 6= y in F , there exists Θx , Θy two open subsets of F , such that, [ Θx ∩ Θy = φ, x ∈ Θx , y ∈ Θy , and Θx Θy = F. Exercise 5. Let K1 , and K2 be compact subsets in Rd1 , and Rd2 , respectively. d1 , d2 are integers. Then, dim(K1 × K2 ) ≥ dimK1 + dimK2 . The equality is not true in general. Exercise 6. (1) Let E ⊂ Rn , and µ be a Borel measure, such that, µ(E) > 0, and that µ(B(x, r)) ≤ Crα , for some constant 0 < C < ∞, and for any Ball B(x, r), x ∈ E. Show that H α (E) ≥ H α (E) ≥ µ(E)/c, and deduce that dim(E) ≥ α. (2) Let µ be a probability measure on A ⊂ [0, 1], satisfying µ(I) ≤ C|I|α , for all interval I ⊂ [0, 1]. Let next F ⊂ [0, 1]2 be such that, dimF ≥ α, and denote Fx = {t ∈ [0, 1]; (x, t) ∈ F }. Show that dim(Fx ) ≤ dim(F ) − α, for µ-almost every x in A. Exercise 7. Let 0 ≤ α ≤ 21 . 1. Plot the set C of points in the complex plane as follows: • Start with the line segment [0, 1] • At each stage replacing each interval I = [x, y] by the union of intervals [x, z] ∪ [z, w] ∪ [w, z] ∪ [z, y], where z=

1 (x + y), and w = z + iα(y − x). 2

page 75

February 7, 2022 16:4

76

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

2. Compute the Hausdorff dimension of the resulting set. 3. Describe the exact nature of the set P for α = 21 . 4. Consider next the Hausdorff distance defined for two compact sets A, B as dH (A, B) = max{max dist(a, B), max dist(b, A)}. a∈A

b∈B

If An → A in the Hausdorff metric, does dim(An ) → dim(A)? Exercise 8. Let E be a Cantor set in Rn . Is the projection of E onto a k-dimensional subspace is necessarily a Cantor set in Rk ? Exercise 9. Let K ⊂ Rn be the Cantor set associated to the similitudes {f1 , . . . , fn }, that satisfy the open set condition. Prove that the set associated to {f1 , . . . , fn−1 } is a subset of strictly smaller dimension. Exercise 10. Let A ⊂ Rd , and B ⊂ Rn be compact sets. a. Show that dim(A) + dim(B) ≤ dim(A × B) ≤ dim(A) + dim(B). b. Give an example of subsets E, and F of R such that dim(E) + dim(F ) ≤ dim(E × F ) < dim(E) + dim(F ) + 1.

page 76

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Chapter 5

Capacity Dimension of Sets

As it is mentioned in the previous chapters, the main problem in computing the Hausdorff dimension of sets appeared directly from its mathematical definition as an outer measure, by means of coverings. This is why researchers in different fields have proposed many variants to estimate the Hausdorff dimension, and/or its modified variants, especially when dealing with applications. One of the well known variants of Hausdorff dimension is the so-called Capacity dimension of sets, which will be studied in the present chapter. Such a dimension has many advantages, and applications. It relates, for example, the theory of Hausdorff, and generally fractal dimensions to physics, as it is strongly related to the physical Capacity of physical instruments such as conductors. For detailed study, and more backgrounds on the concept of Capacity as well as its relation, and application in different fields, the readers may refer to [Beardon (1965); B´elair (1987); Billingsley (1960, 1961, 1965); Boyd (1973); Buck (1970, 1973); Choquet (1961); David and Semmes (1997); Edgar (1998, 2008); Eggleston (1949, 1951); Falconer (1994, 1990); Frostman (1935); Kigami (2001); Pesin (1997); Selezneff (2011)] 5.1

Generalities

Let µ be a probability measure on Rd , and α > 0. Denote Z Z dµ(x)dµ(y) , Iα (µ) = |x − y|α where, for a vector x ∈ Rd , the notation |x| designates its Euclidean norm. We will denote also for a subset A in Rd its diameter by |A| or diam(A), which is evaluated as |A| = diam(A) = sup |x − y|. x,y∈A

77

page 77

February 7, 2022 16:4

ws-book9x6

78

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Definition 5.1. Let A be a compact subset in Rd . A is said to be an αpositive Capacity, and we write Capα A > 0, if there exists a probability measure µ on Rd supported by A, such that, Iα (µ) < ∞. Otherwise, we write Capα A = 0. We immediately have the following characterization. Lemma 5.1. Capα A = 0 ⇒ Capγ A = 0, ∀γ > α. Proof. We have Z Z Z Z Iα (µ) dµ(x)dµ(y) dµ(x)dµ(y) = ≥ . Iγ (µ) = |x − y|γ |x − y|γ−α |x − y|α |A|γ−α As Capα A = 0, we get Iα (µ) = ∞. Hence, Iγ (µ) = ∞. So, Capγ A = 0. Definition 5.2. The Capacity dimension of A is defined by dimc A = inf{α > 0, Capα = 0}. Theorem 5.1. Let A be a compact in Rd , then, dimc A = dimA. Proof. Let 0 < α < γ, and µ be a probability measure on Rd , such that, µ(A) = 1. There exists R > 0, such that, |x − y| ≤ R, ∀x, y ∈ A. Consequently, Z Z Z Z dµ(y)dµ(x) 1 dµ(y) Iα (µ) Iγ (µ) = ≥ γ−α ( )dµ(x) ≥ γ−α . |x − y|γ R |x − y|α R Therefore, it suffices to show that Hγ (A) > 0 ⇒ Capα A > 0 ⇒ Hα (A) > 0. Indeed, it is straightforward that dimA = 0 =⇒ Caps A = 0, ∀s > 0. Consequently, dimc A < s, ∀s > 0, which means that dimc A = 0. Next, for dimA > 0, let γ be such that, 0 < γ < dimA. We immediately obtain Capγ A > 0 =⇒ dimc A ≥ γ. Consequently, dimc A ≥ dimA. Now, let α be such that, 0 < α < dimc A. We have Iα (µ) = ∞ =⇒ dimA ≥ α =⇒ dimA ≥ dimc A.

page 78

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Capacity Dimension of Sets

b4547-main

79

Indeed, there exists a probability measure µ on Rd such that µ(A) = 1, and µ(B) ≤ M |B|γ . Consequently, µ({x}) = 0, ∀ x ∈ A. Now, observe that Z Z Z dµ(y) dµ(y) dµ(y) = + . α |x − y|α |x − y| |x − y|α 0 0, such that, µ(At ) > 0, where Z o n dµ(y) ≤ t . At = x ∈ A, |x − y|α S T Whenever At ⊆ j Bj , with Bj At 6= ∅, we get, for x ∈ At ∪ Bj , |x − y| ≤ |Bj |, ∀y ∈ Bj , and thus, Z dµ(y) µ(Bj ) ≤ ≤ t. |Bj |α |x − y|α Bj As a consequence, 1 µ(Bj ) ≤ |Bj |α , t which yields that X 1 0 < µ(At ) ≤ |Bj |α . t As a result, Hα (At ) > 0.

page 79

February 7, 2022 16:4

80

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

5.2

Self-similar sets

Definition 5.3. A function f : Rd −→ Rd is said to be contractive or a Contraction of Rd if there exists a constant c, 0 < c < 1, such that, |f (x) − f (y)| ≤ c|x − y|; ∀x, y ∈ Rd . In the case of equality we say that f is a self-similar function or a self-similar with ratio c. Consider next a finite set of contractions S = (Si )1≤m on Rd . We define by induction a set of set-valued maps S k (E), k ≥ 0 as follows. For E ⊂ Rd , let n [ S 0 (E) = E, S 1 (E) = S(E) = Si (E), i=1

and S k+1 (E) = S(S k (E)), ∀ k. Definition 5.4. A subset F ⊂ Rd is said to be S-invariant, if it satisfies n [ Si (F ) = F. S(F ) = i=1

The following result deals with the existence, and uniqueness of S-invariant sets . Theorem 5.2. Let S = (Si )1≤i≤m be a finite set of contractions on Rd . There exists a unique non-empty compact F in Rd which is S-invariant. Furthermore, \ F = S k (A), k≥0

for all non-empty compact A in Rd , such that, S(A) ⊂ A. Proof. We will apply the fixed point theorem. Let C be the collection of all compact subsets in Rd , and consider the map S : C −→ C, which maps Sn a compact subset X to its image S(X) defined by S(X) = i=1 Si (X). We shall consider a distance δ on C, for which, • (C, δ) is a complete metric space. • S is contractive on C: S(X), S(Y )) ≤ cδ δ(X, Y ), ∀ X, Y ∈ C, and for some constant cδ , 0 < cδ < 1.

page 80

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Capacity Dimension of Sets

b4547-main

81

To do it consider the Hausdorff distance δ defined as follows. For X ∈ C, and a > 0, denote n o [ B(X, a) = y ∈ Rd ; d(y, X) ≤ a = B 0 (x, a). x∈X

It is straightforward that B(X, a) is a compact subset of Rd , and thus, B(X, a) ∈ C. The distance δ will be defined by n o δ(X, Y ) = inf a > 0; X ⊂ B(Y, a), and Y ⊂ B(X, a) . Next, we shall show that (C, δ) is complete. Let (Xn )n≥1 be a Cauchy sequence in (C, δ). For all  > 0, there exists p ∈ N, such that, δ(Xn , Xm ) < , ∀m ≥ n ≥ p. Hence, the set

S∞

i=1

Xi is bounded, due to the fact that Xn ⊂ B(Xp , ), ∀n ≥ p.

Denote next X=

\ [ ( Xj ). i

j≥i

It is straightforward that X is a non-empty compact subset of Rd , and thus, X ∈ C. Moreover, lim Xn = X.

n,→+∞

Indeed, for  > 0, and p defined previously, we have Xj ⊂ B(Xn , ), ∀j ≥ n ≥ p. As a result, [

Xj ⊂ B 0 (Xn , ), ∀j ≥ n ≥ p.

j≥i

Consequently, X ⊂ B(Xn , ). Conversely, let x ∈ Xn , n ≥ p. We immediately observe that Xn ⊂ B(Xk , ), ∀k ≥ n ≥ p. Therefore, for k ≥ n ≥ p, there exists yk ∈ Xk , such that, d(x, yk ) ≤ . S Observe also that yk ∈ j≥k Xj . There exists consequently a sub-sequence

page 81

February 7, 2022 16:4

ws-book9x6

82

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

(ynk )k convergent to y in X, and satisfying, d(x, ynk ) ≤ , which means that d(x, y) ≤ . So, d(x, X) ≤ , which yields that x ∈ B(X, ), and thus, Xn ⊂ B(X, ). Consequently, δ(Xn , X) ≤ , ∀n ≥ p. As a result, we conclude that (ϕ, δ) is complete. It remains to show that S is a Contraction relatively to the distance δ. We will show precisely that δ(S(X), S(Y )) ≤ sup δ(Si (X), Si (Y )). 1≤i≤m

Let a > sup δ(Si (X), Si (Y )). Then, ∀ i, 1 ≤ i ≤ m :, we have 1≤i≤m

Si (X) ⊂ B(

m [

Sj (Y ), a) , and Si (Y ) ⊂ B(

j=1

m [

j=1

Consequently, [

Si (X) ⊂ B(

m [

Sj (Y ), a),

j=1

1≤i≤m

and [

Si (Y ) ⊂ B(

m [

Sj (X), a).

j=1

1≤i≤m

Hence, δ

 [

Si (X),

1≤i≤m

[

 Si (Y ) ≤ a.

1≤i≤m

This is true ∀a > sup δ(Si (X), Si (Y )). 1≤i≤m

So, equation (5.1) holds. Now, as we already have δ(Si (X), Si (Y )) ≤ ci δ(X, Y ), ∀ i,

Sj (X), a).

(5.1)

page 82

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Capacity Dimension of Sets

where ci is the ratio of the Contraction Si , we take Cδ =

83

sup ci . We 1≤i≤m

thus, conclude that S is contractive on C. So, finally, there exists a unique compact subset F of Rd , such that, S(F ) = F =

m [

Si (F ).

i=1

It remains now to prove the last point in the theorem. To do it, we will show some more general result. Let (Xk )k be a decreasing sequence in C, and denote \ X = lim Xk = Xk . k→∞

k

We claim that δ(Xk , X) is decreasing to 0 in R. Indeed, the sequence (δ(Xk , X))k is decreasing. If it is bounded away from 0, there exists a > 0, such that, δ(Xk , X) ≥ a > 0, ∀k. This yields that Xk * B(X, a), ∀k. Hence, ∀k, there exists xk (∈ Xk ), such that, d(xk , X) ≥ a. On the other hand, there exists a sub-sequence (xnk )k convergent to some point x ∈ X. Consequently, lim d(xnk , X) = d(x, X) = 0 ≥ a > 0,

n→∞

which is contradictory. As a result, δ(Xk , X) → 0, as k → ∞, which means that (Xk )k converges to X in (C, δ). Now, for Xk = S k (A), we get a decreasing sequence, which by the previous result, converges to \ \ X= Xk = S k (A). k

Hence, F =

T

k≥0

S k (A).

k

page 83

February 7, 2022 16:4

84

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Theorem 5.3. Let S = (Si )1≤i≤m be a set of similarities with respective ratios (ci )1≤i≤m . Let F be the unique non-empty compact S-invariant. Assume that S satisfies the open set condition: There exists a non-empty open, and bounded set V in Rd , such that m [ \ S(V ) = Si (V ) ⊂ V and Si (V ) Sj (V ) = φ, ∀ i 6= j. (5.2) i=1

Then, dimF = s, where s is the unique solution of

n X

csi = 1. Besides, we

i=1

have 0 < Hs (F ) < ∞. Example 5.1. We propose to compute the Hausdorff, and the Capacity dimension of the triadic Cantor set, by applying Theorem 5.3. We consider the set S = (S1 , S2 ), where the contractions S1 , and S2 are defined on R by 1 2 1 S1 (x) = x and S2 (x) = x + . 3 3 3 Let F be the unique non-empty compact S-invariant. We claim that F = C, the triadic Cantor set. To do this, consider A = [0, 1]. We have 1 2 S(A) = [0, ] ∪ [ , 1] ⊂ A. 3 3 Consequently, \ F = S k (A). k≥1

It remains to show that S k (A) = Ek , ∀k. For k = 0, we have S 0 (A) = A = [0, 1] = E0 . For k = 1, 1 [2 S 1 (A) = [0, ] [ , 1] = E1 . 3 3 k So, whenever for a step k, S (A) = Ek , we get S k (A) = Ek =

[

(i1 ,...,ik

=

Si1 ◦ · · · ◦ Sik (A)

)∈{1,2}k

[ (i1 ,...,ik )∈{1,2}k

Ii1 ,...,ik ,

page 84

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Capacity Dimension of Sets

85

where Ii1 ,...,ik are the triadic intervals composing Ek . This yields that   [ S k+1 (A) = S1 (Ii1 ,...,ik ) ∪ S2 (Ii1 ,...,ik ) . (i1 ,...,ik )∈{1,2}k

Otherwise, S k+1 (A) =

[

Ii1 ,...,ik ,ik+1 = Ek+1 .

(i1 ,...,ik ,ik+1 )∈{1,2}k

Finally, to compute the Hausdorff dimension, we chose V =]0, 1[. We immediately have \ S(V ) ⊂ V , and S1 (V ) S2 (V ) = φ. So, dimC = s, where s is the unique solution of cs1 + cs2 = 1, or equivalently, 2

 1 s 3

= 1,

which yields that s=

log 2 . log 3

Theorem 5.4. Let S = (Si )1≤i≤m be a set of contractive self-similarities, with respective ratios (ci )1≤i≤m . Assume further that S satisfies the open set condition (5.2). Then, there exists a Borel probability measure µ on Rd , supported by F , such that, ∀X a Borel subset of Rd , we have m X

µ(X) =

csi µ(Si−1 (X)),

i=1

where s is the unique solution of m X

csi = 1.

i=1

Proof. Let x ∈ F be fixed, and denote for k ∈ N, the linear form Lk : Cc (Rd ) → R, such that, X Lk (f ) = csi1 ...csik f (xi1 ,...,ik ), (i1 ,...,ik )∈{1,...,m}k

page 85

February 7, 2022 16:4

ws-book9x6

86

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

where xi1 ...ik = Si1 ◦ ...Sik (x). It holds that (Lk (f ))k is a Cauchy sequence in R. Let L be the linear form limit. Riesz theorem yields that there exists a Borel probability measure µ on Rd , such that, Z L(f ) = f dµ. Standard techniques permit to conclude that µ(Rd ) = µ(F ) = 1. On the other hand, we have Z Z m X f dµ = csi f ◦ Si dµ, ∀f ∈ Cc (Rd ). i=1

Consequently, by the density of Cc (Rd ) in the Lp spaces, we get Z Z m X f dµ = csi f ◦ Si dµ, i=1

for any integrable function f . So, for f ≡ 1X , we obtain µ(X) =

m X

csi µ(Si−1 (X)).

i=1

Lemma 5.2. Let (Vi )i∈I be a family of disjoint open sets in Rd . Let 0 < r1 < r2 , ρ > 0, be such that, ∀i, Vi contains a Ball of radius r1 ρ, and is contained in a Ball of radius r2 ρ. Then, every Ball of radius ρ, intercepts, at most (1 + 2r2 )d r1−d elements of the family (Vi )i∈I . Proof of Theorem 5.3. It suffices to prove that 0 < H s (F ) < +∞. Assume that Jk = {(i1 , ..., ik ), 1 ≤ ij ≤ m}. For A ⊂ Rd , let Ai1 ,...,ik = Si1 ◦ ... ◦ Sik (A). We have F =

m [ i=1

Si (F ) =

[ (i1 ,...,ik )∈Jk

Fi1 ...ik .

page 86

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Capacity Dimension of Sets

87

Moreover, |Fi1 ...ik | = ci1 ...cik |F |. So that, X

|Fi1 ...ik |s =

X

i

csi1 ...csik |F |s = |F |s .

i

s

Hence, H (F ) < +∞. Theorem 5.4 implies that there exists a Borel probability measure µ on Rd , carried by F , such that, for any Borel set X in Rd , we have µ(X) =

m X

csi µ(Si−1 )(X).

i=1

We claim that µ is s-H¨ olderian. Indeed, let 0 < ρ < min ci . We shall prove i

that, for any Ball B(., ρ), we have µ(B(., ρ)) ≤ C|B(., ρ)|s , where C > 0 is a constant. Denote next µi1 ...ik (X) = µ((Si1 ◦ ... ◦ Sik )−1 (X)). Hence, µi1 ...ik is a probability measure on Rd . Si being bijective, so µi1 ...ik (Fi1 ...ik ) = 1, and Fi1 ...ik ⊂ V i1 ...ik . Now, observe that µ=

X

csi µi = ... =

X

csi1 ...csik µi1 ...ik ,

and let η1 < η2 be such that, B(., η1 ) ⊂ V ⊂ B(., η2 ). Then, B(., ci1 , ..., cik η1 ) ⊂ Vi1 ...ik ⊂ B(., ci1 , ..., cik η2 ). Let next k0 ∈ N be such that, (max ci )k0 < ρ min ci , and let k be the smallest integer, for which, (i1 , ..., ik0 ) ∈ Jk0 , 1 < k < k0 , and (min ci )ρ ≤ ci1 ...cik ≤ ρ. We define the mapping Φ : Jk0 → Φ(Jk0 ), by Φ((i1 , ..., ik0 )) = (i1 , ..., ik ).

(5.3)

page 87

February 7, 2022 16:4

ws-book9x6

88

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

We get X

µ=

csi1 ...csik µi1 ,...,ik0

(i1 ...ik0 )∈Jk0

X

=

X

csi1 ...csik µi1 ...ik0 . 0

(j1 ...jk )∈Φ(Jk0 ) (i1 ,...,ik0 )∈J0

Observing that Φ−1 (j1 , .., jk ) = {(j1 , .., jk , lk+1 , .., lk0 )/lj ∈ {1, ..., m}}, we obtain X X cslk+1 ...cslk µik+1 ...lk0 µ= csj1 ...csjk 0

=

lj (j1 ,...,jk ) X s s cj1 ...cjk µj1 ...jk .

Otherwise, (min ci )ρ ≤ cj1 ...cjk ≤ ρ. So that, (min ci )ηi ρ ≤ cj1 ...cjk η1 . As a result, B(., (min ci )η1 ρ) ⊂ B(., cj1 ...cjk η1 ) ⊂ Vj1 ...jk ⊂ B(., cj1 ...cjk η2 ) ⊂ B(., η2 ρ). On the other hand, (min ci )ρ ≤ cj1 ...cjk ≤ ρ, which yields that (min ci )η1 ρ ≤ cj1 ...cjk η1 . Hence, B(., (min ci )η1 ρ) ⊂ B(., cj1 ...cjk η1 ) ⊂ Vj1 ...jk ⊂ B(., cj1 ...cjk η2 ) ⊂ B(., η2 ρ). Then, Lemma 5.2, with r1 = (min ci )η1 , the family (Vj1 ...jk ), and r2 = η2 , yields that, B(., ρ) meets at most N = (1 + 2r2 )d r1−d elements of the family (V j1 ...jk ). Thus, X µ(B(., ρ)) = csj1 ...csjk µ(j1 ,...,jk ) (B(., ρ)) (j1 ,...,jk )

≤ N ρs N = s |B(., ρ)|s . 2 So, µ is s-H¨ olderian.

page 88

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Capacity Dimension of Sets

b4547-main

89

Proof of Lemma 5.2. Whenever, we have R ≥ ρ(X + 2r2 ), and N Cd (r1 ρ)d ≤ Cd (1 + 2r2 )d ρd , we immediately deduce that N ≤ (1 + 2r2 )d r1−d .

Proof of Lemma 5.4. Let x ∈ F be fixed. For k ∈ N, consider the linear form Lk defined previously, and fix f ∈ Cc (Rd ). We have, ∀k, p ∈ N,   X Lk (f ) − Lp (f ) = (ci1 ...cip )s f (xi1 ...ik ) − f (xi1 ...ip ) . Now, observing that (f (xi1 ...ik ) , f (xi1 ...ip )) ∈ Fi21 ...ik . We obtain |Fi1 ...ik | = ci1 ...cik |F |. Next, as f is uniformly continuous, then, (Lk (f ))k is a Cauchy sequence in R. Denote L(f ) = lim Lk (f ). k→+∞

It follows that L is a linear positive form. Therefore, there exists a Borel measure µ ≥ 0 on M, a σ-algebra on Rd , such that, Z L(f ) = f dµ f ∈ Cc (Rd ). Rd

In particular, there exists (fn )n ⊂ Cc (Rd ) Increasing to f = χRd , and for n large enough, F ⊂ B(0, n). On the other hand, using Lk (fn ), we get for n >> 1, Z lim fn dµ = 1, k→+∞

and Z

f dµ = µ((Rd )) = 1.

We now claim that µ(F ) = 1 (⇐⇒ µ(F c ) = 0).

page 89

February 7, 2022 16:4

ws-book9x6

90

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Indeed, let f ≡ 1 on K, support(f ) ⊂ Fc , and f ∈ Cc ((Rd )). We get Z µ(K) ≤ f dµ = lim Lk (f ) = 0, ∀K ⊂ F c . k→+∞

It results that sup µ(K) = µ(F c ) = 0 ⇒ µ(F ) = 1. K⊂F c

Next, observing that, for k ≥ 2, f (xi1 ...ik ) = f ◦ Si1 (xi2 ...ik ), we get Lk (f ) =

m X

csi Lk−1 (f ◦ Si ).

i=1

Letting k → ∞, we obtain Z X Z f dµ = csi f ◦ Si dµ, ∀f ∈ Cc (Rd ). Using the density of Cc (Rd ), we obtain the desired result. Proposition 5.1. Let N ∈ M be such that, µ(N ) = 0, then Sj−1 (N ) ∈ M , and µ(Sj−1 (N )) = 0. Proof. Let fn → 1X , and µ(N ) = 0. Hence, fn ◦ Si −→ 1X ◦ Si , µ almost everywhere. Now, notice that |fn ◦ Si | ≤ 1, and Si−1 (CR Nd ) = CR Si−1 (N ). Let V be a bounded open set, with µ(Sj−1 (V )) ≤

1 µ(V ), csj

and consider 1 }. n So, Vn is open, and Vn is compact ⊂ V . Let next f ∈ Cc (Rd ) be such that, 0 ≤ f ≤ 1, f ≡ 1 on Vn , and support(f ) ⊂ V . We get Z Z −1 s s cj µ(Sj (Vn )) ≤ cj f ◦ Sj dµ ≤ f dµ = µ(V ). Vn = {x ∈ V, d(x, ∂V ) >

Now, remark that Vn ↑ V =

[ n

Vn , and Sj−1 (V ) =

[ n

Sj−1 (Vn ).

page 90

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Capacity Dimension of Sets

91

So, for arbitrary open set V , we chose Vn = V ∩ B(0, n), and we use the result proved previously. Let next N ∈ M be such that, µ(N ) = 0. There Tn exists Vn open, such that, N ⊂ Vn , and µ(Vn ) ≤ n1 . Denote Wn = k=1 Vk . So, Wn is open, ∞ \ Wn ↓ W = Wn , n=1

and µ(Wn ) ≤

1 n.

We have

Sj−1 (N ) ⊂ Sj−1 (W ) =

∞ \

Sj−1 (Wn ) ∈ B ⊂ M,

n=1

and µ(Sj−1 (Wn )) ≤

1 1 . csj n

Consequently, µ(Sj−1 (W )) = 0, which implies that µ(Sj−1 (N )) = 0.

5.3

Billingsley dimension

Let c ∈ N, c ≥ 2, and F be the family of c-adic intervals [ ckn , k+1 cn [, where k, n ∈ N. Let next µ be a non-atomic probability measure on Rd . For E ⊂ Rd , α ≥ 0, and  > 0, denote nX o [ α Hµ, (E) = inf µ(Ij )α , E ⊂ Ij , Ij ∈ F, µ(Ij ) ≤  . j

j

As for the case of the Hausdorff measure, we obtain here an Increasing function in . So, denote α Hµα (E) = lim Hµ, (E). →0

We get here an outer metric measure on Rd . Furthermore, there exists a cut-off value D ∈ R such that Hµα (E) = ∞, for α < D,

and Hµα (E) = 0, for α > D.

We write dimµ E = D, and call it the µ-dimension of E. Proposition 5.2. The following assertions are true.

page 91

February 7, 2022 16:4

ws-book9x6

92

(1) (2) (3) (4) (5)

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

0 ≤ dimµ E ≤ 1. dimµ {a} = 0, ∀a ∈ R. S dimµ n En = supn dimµ En . Whenever E is a Borel set, and µ(E) > 0, then, dimµ E = 1. For E ⊂ F , we have dimµ E ≤ dimµ F .

Proof. (1) From its definition, we see that dimµ E ≥ 0. Let next s > 1, and ε > 0. For any ε-covering (Ij )j ⊂ F of E, we have [  X s Hµ,ε (E) ≤ µ(Ij ) ≤ µ Ij ≤ 1. j

j

Consequently, Hµs (E) ≤ 1, ∀ s > 1, which means that dimµ E ≤ s, ∀ s > 1. Hence, dimµ E ≤ 1. (2) For all n ∈ N, the singleton {a} is contained in one interval Ik ∈ F, (for which ckn ≤ a < k+1 cn ). As a result, for all ε > 0, and all n large enough log ε (n > − log c ), we get α Hµ,ε ({a}) ≤

1 , cnα

which yields that α Hµ, ({a}) < ∞, ∀ α ≥ 0.

Consequently, dimµ ({a}) ≤ α, ∀ α ≥ 0. With assertion 1, we get dimµ ({a}) = 0. (3) It is obvious that any covering of F is a covering of E. Therefore, ∀α > dimµ F , we get α α Hµ,ε (E) =≤ Hµ,ε (F ), ∀ ε > 0.

As a result, Hµα (E) ≤ Hµα (E) = 0. This yields that dimµ E ≤ α, ∀ α > dimµ F. Consequently, dimµ E ≤ dimµ F . (4) Let α > sup dimµ Ep . It holds from that p

[ X 0 ≤ Hµα ( Ep ) ≤ Hµα (Ep ) = 0. p

p

page 92

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Capacity Dimension of Sets

93

Consequently, Hµα (

[

Ep ) = 0,

p

which yields that dimµ (

[

Ep ) ≤ α, ∀ α > sup dimµ Ep . p

p

As a result, [ dimµ ( Ep ) ≤ sup dimµ Ep . p

p

The converse follows from assertion 3. (5) Let α < 1. For all ε > 0, and (Ij )j ⊂ F and ε-covering of E, we have X X µ(E) ≤ µ(Ij ) ≤ µ(Ij )α . j

j

Therefore, α 0 < µ(E) ≤ Hµ,ε (E).

Letting ε ↓ 0, we obtain Hµα (E) > 0, ∀ α < 1. As a results, dimµ E ≥ 1. By assertion 1, we deduce the equality desired. Theorem 5.5. For t ∈ R, let In (t) be the c-adic interval of order n, which contains t. Let µ, ν be two non-atomic probability measures on R, and α ≥ 0. Then, for any subset o n log µ(In (t)) =α , E ⊂ t ∈ R, lim n→+∞ log ν(In (t)) we have dimν E = αdimµ E. Proof. Assume that 0 < α < +∞. We shall show, in a first step, that for E ⊂ {t ∈ R, limn

log µ(In (t)) ≥ α}, log ν(In (t))

we have dimν E ≥ αdimµ E.

page 93

February 7, 2022 16:4

ws-book9x6

94

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

T So, assume that, ∀Ij ∈ F, such that, Ij E 6= ∅, we have ν(Ij ) 6= 0. Let next β > dimν E, and η > α1 . We claim that dimµ E ≤ ηβ. Indeed, let 1 Ek = {t ∈ E, ∀n, ν(In (t)) ≥ , or µ(In (H)) ≤ ν(In (H))}. k S Hence, (Ek )k % E = k Ek . Indeed, t ∈ E, there exists p, such that, log µ(In (t)) 1 ≥ , ∀n ≥ p. log ν(In (t)) η 1 Let next k be such that, ≤ ν(Ip (t)). We get t ∈ Ek . Therefore, it suffices k to prove that dimµ Em ≤ ηβ, ∀m. Consider a rectangle (Ij )j of Em , such that, ∀j, Ij ∈ F, and ν(IJ ) ≤ 
0 be such that, dimν E ≤ η. We notice immediately that n o n log µ(In (t)) E ⊂ t ∈ R, lim ≤η . log ν(In (t)) Whenever Ij ∩ E 6= ∅ satisfies µ(Ij ) 6= 0, we get n o 1 Ek = t ∈ R, µ(In (t)) ≥ , or ν(In (t))η ≤ µ(In (t)) . k S So, the same techniques as above yield that E = k Ek , and consequently, dimν Ek ≤ η. 5.4

Eggleston theorem

Let c ≥ 2 in N, denote Nc = {0, 1, · · · , c − 1}, and Nc∗ the set of finite words constructed with Nc as an alphabet. For j ∈ Nc∗ , j = j1 j2 , . . . , jn , with jk ∈ Nc , we write |j| = n the length of the word j. For k ∈ Nc , let Nk (j) be the number of appearance of the letter k in j, and n n X jk X jk 1 Ij = , + n . ck ck c k=1

k=1

It is clear that Ij is a c-c-adic interval ⊂ [0, 1[. Finally, for k ∈ Nc , and t ∈ [0, 1[ let ϕkn (t) = Nk (j), where t ∈ Ij , |j| = n. Theorem 5.6. Let p = (pk )0≤k≤c−1 be a probability vector, and n o 1 E = t ∈ [0, 1[, lim ϕn (t) = pk , ∀k ∈ Nc . n→+∞ n Then, n 1 X dimE = − pk log pk . log c k=0

To prove this theorem, we need some preliminary results from probability theory. Theorem 5.7. Let (Xn )n be a sequence of real random variables defined on a probability space (Ω, Q, P), such that, the Xn areR independent, and with the same law, and (Xn )n ∈ L2 . Let M = E(X1 ) = Ω X1 dP. Then n 1X lim Sn = lim E(Xi ) −→ M, a.e. n→+∞ n→+∞ n i=1

page 95

February 7, 2022 16:4

96

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Proof. It follows from the equality n X
V ar(X1 )
1 Xi = . (E) (Sn − M )2 = 2 V ar n n i=1

Proof of Theorem 5.6. We take in Billingsley’s theorem ν = m the Lebesgue measure, and so, the problem is to look for µ, such that, n o log µ(In (t)) E ⊂ t ∈ [0, 1[ ; −→ D , log |In | where n

D=−

1 X pk log pk . log c k=0

We next consider (R, BR ), and assume that 0 < pk < 1. Let µn be the measure on R, such that, µ − n(Ij1 ...jn ) = pj1 ...pjn . Such a measure exists. Indeed, take the density measure relatively to the Lebesgue one on I, defined by pj ...pjn µn (Ij1 ...jn ) = ( 1 )m(Ij1 ...jn ). m(Ij1 ...jn ) We get immediately, µn ([0, 1[) = 1. Hence, it suffices to prove that there exists a sub-sequence (µnk )k convergent to µ, which is equivalent to µnk (Ij1 ...jn ) →k µ(Ij1 ...jn ), and thus, µ(∂Ij1 ...jn ) = 0. Indeed, notice that (µn (Ij1 ...jn ))n is a bounded sequence on R. So, there exists a sub-sequence (µnk )k , such that, (µnk (Ij1 ...jn ))k is convergent in R. So, denote µ(Ij1 ...jn ) its limit. The weak convergence of measures serves to conclude that µ is also a measure. Moreover, µ({α}) ≤ limk tnk (] . . . [) ≤ 2pn+s (s→∞) −→ 0, where p, n, s are, such that, In (s) 3 α. Now observe that n o log µ(In (t)) E ⊂ t ∈ [0, 1[, lim =D , n→∞ log |In (t)|

page 96

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Capacity Dimension of Sets

97

and that N (j)

µ(In (t)) = p0 0

N

c−1 . . . pc−1

(j)

=

c−1 Y

ϕk (t)

pk n

,

k=0

where j is, such that, In (t) = Ij1 ...jn . Consequently, c−1

c−1

k=0

k=0

1 X k 1 X1 k log µ(In (t)) =− ϕn (t) log pk = − ϕ (t) log pk . log |In (t)| n log c log c n n Next, consider

ψnk (t)

= δjn ,k . It holds easily that n

1 k 1X k ϕn (t) = ψ (t) −→ pk . n−→∞ n n s=1 s

5.5

Exercises for Chapter 5

Exercise 1. Show that δ defined in section is a distance on F. Exercise 2. Consider the set S = (S1 , S2 ) where the contractions S1 , and S2 are defined on R by 1 2 1 x and S2 (x) = x + . 3 3 3 Let F be the unique non-empty compact S-invariant. S1 (x) =

(1) Show that F = C, the triadic Cantor set. (2) Compute the Hausdorff dimension of F using Theorem 5.3. Exercise 3. Give an example to show that α-Capacity is not additive. Exercise 4. Show that if {Kn } is an increasing sequence of sets, that is, Kn ⊂ Kn+1 , for all n ∈ N, then Capα (∪n Kn ) = lim Capα (Kn ). n

Exercise 5. Prove that, for all E, F subsets of Rd , we have Capα (E ∪ F ) ≤ Capα (E) + Capα (F ).

page 97

February 7, 2022 16:4

98

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Exercise 6. Consider K = {0} ∪ {1, 21 , 13 , 14 , ...}. Show that dimM (K) =

1 . 2

Exercise 7. For 0 < α, β < 1, let Kα,β be the Cantor set obtained as an intersection of 0 1 the following nested compact sets. Kα,β = [0, 1]. The set Kα,β is obtained by leaving the first interval of length α, and the last interval of length β, and n−1 n removing the interval in between. To get Kα,β , for each interval I in Kα,β , leave the first interval of length α|I|, and the last interval of length β|I|, and remove the sub-interval in between. Compute the Minkowski dimension of Kα,β . Exercise 8. Develop a proof of Theorem 5.3. Exercise 9. Consider the kernel K(x, y) =

|y|d−2 G(x, y) = , G(0, y) |x − y|d−2

for x 6= y in Rd , and K(x, x) = ∞. Let F be any closed set in Rd ; d ≥ 3, and denote !−1 Z Z CopK (F ) = inf K(x, y)dν(x)dν(y) . ν(F )=1

F

F

Consider next the spherical shell FR = {x ∈ Rd : 1 ≤ |x| ≤ R}. Show that limR∞ CapK (FR ) = 2. Exercise 10. For a kernel K, we denote CapK the associated Capacity, and Cap∞ K the asymptotic Capacity defined for a set A by Cap∞ K (A) =

{X

inf CapK (A \ X). finite}

For a cube C = [a1 , an+1 ] × ... × [ad , an+d ], n ≥ 1, denote d(C) the distance of the farthest point in C to 0, and |C| the diameter of C. For A ∈ Zd , define X |Cj | α )α , HD (A) = inf ( d(C ) J j

page 98

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Capacity Dimension of Sets

b4547-main

99

where the inf is over all coverings of A by cubes. Define next the discrete Hausdorff dimension of A, by α dimD (A) = inf{α > 0; HD (A) < ∞}.

Consider next, for α > 0, the kernel Kα (x, y) =

|y|α . 1 + |x − y|α

1. Show that for all A ⊂ Zd , and for all α > β > 0, we have α CapKα (A) ≥ CHD (A),

where C > 0 is a positive number depending only on α, and β. α 2. Show that if CapKα ∞ (A) > 0, then, HD (A) = ∞. ∞ 3. Show that dimD (A) = inf{α : CapKα (A) = 0}.

page 99

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Chapter 6

Packing Measure and Dimension

The packing measure, and dimension constitute the second original essays in fractal analysis, and geometry. This chapter constitutes therefore a second essential part of the book. In which, we focus on the notion of packing measure, which is also in the heart of fractal analysis, and geometry. In fractal analysis, we sometimes say that a set, and/or a measure is fractal when its Hausdorff dimension differs from its packing one. In the present chapter, we will provide in details the original construction of such a measure, its different variants, and the concept of the packing dimension associated to it. Besides, we discuss also the concept of Box dimension, called sometimes the Bouligand–Minkowski dimension, and its link with Hausdorff, and especially packing dimension. More information, examples, and related topics may be found in [Batakis and Heurteaux (2002); Beardon (1965); Ben Mabrouk and Aouidi (2011); Ben Nasr (1994); Ben Nasr, Bhouri and Heurteaux (2002); Billingsley (1960, 1961, 1965); Boyd (1973); Brown et al (1992); Buck (1970, 1973); Choquet (1961); David and Semmes (1997); Edgar (1998, 2008); Eggleston (1949, 1951); Falconer (1985, 1994, 1990); Frostman (1935); Gervais (2009); Heurteaux (1998); Hutchinson (1981); King (1995); Kigami (2001); Lambert (2012); Mignot (1998); Ngai (1997); Olsen (1995); Peitgen et al (1992a,b); Pesin (1997); Peyriere (1992); Rand (1989); Riedi (1995); Robert (2020); Rogers (1970); Spear (1992); Yeh (2014)].

100

page 100

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Packing Measure and Dimension

6.1

101

Bouligand–Minkowski dimension

For a bounded subset E in Rd , and  > 0, denote N (E) the minimum number of closed Balls of diameter  which cover E, and ∆(E) = lim sup →0

log N (E) . − log 

Denote also n o E = x ∈ Rd , d(x, E) ≤  , where d is the Euclidean distance on Rd . We call -packing of E, a Countable collection of open Balls (Bj )j∈N , centred in E, disjoint, and satisfying also |Bj | ≤ , ∀j, where the symbol |.| stands for the diameter. Let M (E) be the maximum number of Balls, with diameter , constituting an -packing of E. For n ∈ N, let Wn (E) be the number of dyadic cubes of order n which intersect E. Proposition 6.1. For any bounded subset E ⊂ Rd , we have ∆(E) = lim sup →0

log M (E) log Wn (E) log m(E ) = lim sup = lim sup(d− ), →0 →0 − log  − log  log 

where m is the Lebesgue measure. Proof. Denote ∆1 , ∆2 , and ∆3 the second, third, and fourth quantities, respectively. Let  (B(xj , ))j , 1 ≤ j ≤ M (E), 2 be an -packing of E, such that, [  B(xj , ) ⊂ E . 2 j We immediately obtain M (E)Kd ≤ m(E ), for some constant K > 0. Consequently, ∆1 ≤ ∆3 .  Observe now that ∀x ∈ E, there exists a Ball B(xj , ), such that, 2  d(x, Bj ) ≤ . 2

page 101

February 7, 2022 16:4

ws-book9x6

102

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Hence, E⊂

[

B(xj , ),

j

(the closed Balls). This implies that N (E) ≤ M (E), which yields that ∆ ≤ ∆1 . Consider now closed Balls (Bj )j , such that, [ [ E⊂ Bj , and E ⊂ B(xj , 2), j

and that,  (B(xj , ))j , 1 ≤ j ≤ M (E). 2 We thus have m(E ) ≤ N (E)d K 0 , which yields that ∆3 ≤ ∆. Consider now the cubes (Cj )j of order n, such that, Cj Bj be a Ball with the same center of Cj , and satisfying √ d |Bj | = |Cj | = n . 2 We have immediately, [ E⊂ Bj .

T

E 6= ∅. Let also

j

Consequently,

√ N

√

d 2n

(E) ≤ Wn (E) , and

d

2n

√

0 we have ∆(E) = d. ∆(E) = ∆(E). S ∆( n En ) 6= sup∆(En ). n

Proof. (1) Let s > ∆(E), and ε > 0. Let also (B(xi , ri ))1≤i≤Nε (E) be an ε-covering of E with closed balls. We have X Hεs (E) ≤ (2ri )s ≤ (2ε)s Nε (E). i

As s > ∆(E), we deduce that there exists ε0 > 0, such that, ∀ 0 < ε < ε0 , we have log Nη (E) < s. sup − log η 0 0. Consequently, ∆(E) ≤ ∆(F ). (4) Let R > 0 be such that E ⊂ B(0, R), and ε > 0. It holds that εd Nε (E) ≤ Cd Rd ,

page 103

February 7, 2022 16:4

ws-book9x6

104

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

with some constant Cd > 0 depending only on d. This yields that log Cd,R log Nε (E) ≤d+ , − log ε log ε with some constant Cd,R > 0 depending only on d and R. As a consequence, we obtain ∆(E) ≤ d. The positivity follows from the definition of ∆(E). (5) We notice easily, for all ε > 0, that Nε (E)

m(E) ≤

X

m(B(xi , ri )) ≤ Cd Nε (E)εd .

i=1

Therefore, d−

log Cd log Nε (E) log m(E) ≤ + , log ε − log ε log ε

which means that ∆(E) ≥ d. By assertion 4, we get the equality. (6) We obviously have ∆(E) ≤ ∆(E). So, we shall prove the opposite inequality. Let ε > 0. We get Nε (E)

E⊂

[

B(xi , 2ri ).

i=1

Therefore, N2ε (E) ≤ Nε (E), which yields that ∆(E) ≥ ∆(E). (7) It suffices to take E = Q ∩ [0, 1]. 6.2

Packing measure

For E ⊂ Rd , α > 0, and  > 0 denote Pα (E) = sup

X

|Bj |α ,

j

where the upper bound is taken over all -packings (Bj )j of E. Recall that a Countable family of open Balls (Bj )j is said to be an -packing of E if it T satisfies |Bj | ≤ , ∀ j, and Bj Bk = ∅, ∀j 6= k. Denote next P α (E) = lim Pα (E). →0

We have the following properties.

page 104

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Packing Measure and Dimension

105

Proposition 6.3. (1) If E is not bounded, then, P α (E) = +∞, ∀α. (2) If E is bounded, the, P d (E) < +∞. (3) If E ⊂ F , then, P α (E) ≤ P α (F ), ∀α. (4) For all ∀E, F ⊂ Rd , we have [ P α (E F ) ≤ P α (E) + P α (F ). (5) ∀E, F , such that, d(E, F ) > 0, we have [ P α (E F ) = P α (E) + P α (F ). (6) For all subset E ⊂ Rd , the map α 7→ P α (E) satisfies P α (E) < ∞ ⇒ P β (E) = 0, ∀β > α. i+1 i Proof. (1) Let ε > 0, and consider the cubes Ci,n = [ 1/α , 1/α ], i ∈ N. n n We immediately obtain X1 Pεα (E) ≥ = ∞. n n Hence, P α (E) = ∞. (2) Let R > 0 be fixed such that E ⊂ B(0, R). For any Nε > 0, and (B(xi , ri )), 1 ≤ i ≤ Nε (E) and ε-packing of E. We have X (2ε)d ≤ Nε (E). i

On the other hand, (2ε)d Nε (E) ≤ (2R)d . Consequently, P d (E) ≤ (2R)d < ∞. (3) It follows from the fact that any packing of E is obviously a packing for F . (4) Let for ε > 0, (B(xi , ri ))i be an ε-packing of E ∪ F , and denote I = {i B(xi , ri ) ∩ F = ∅}, and J = {i B(xi , ri ) ∩ E = ∅}.

page 105

February 7, 2022 16:4

ws-book9x6

106

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

It holds, therefore, that (B(xi , ri ))i∈I is an ε-packing of E, and (B(xi , ri ))i∈J is an ε-packing of F . As a result, X X X (2ri )α = (2ri )α + (2ri )α ≤ Pεα (E) + Pεα (F ), i

i∈I

i∈J

which yields that P α (E ∪ F ) ≤ P α (E) + P α (E). ) . Let also (B(xi , ri ))i∈I be an ε-packing of E, (5) Let 0 < ε < d(E,F 2 and (B(xi , ri ))i∈J an ε-packing of F . It holds that (B(xi , ri ))i∈I∪J is an ε-packing of E ∪ F . Therefore, X X ≤ (2ri )α + ≤ (2ri )α ≤ Pεα (E ∪ F ), i∈I

i∈J

which yields that P α (E) + P α (E) ≤ P α (E ∪ F ). By assertion 4, we get the equality. (6) For any ε-packing (B(xi , ri ))i of E, and any β > α, we have X X ≤ (2ri )β ≤ (2ε)β−α ≤ (2ri )α . i

i

As a consequence, Pεβ (E) ≤ (2ε)β−α Pεα (E). As P α (E) < ∞, we get P β (E) = 0. Proposition 6.4. ∀E ⊂ Rd , there exists a unique constant D ∈ R, such that, P α (E) = ∞, ∀α < D, and P α (E) = 0, ∀α > D. Furthermore, if E is bounded, we have ∆(E) = D. Proof. Whenever E is not bounded, we put D = +∞. For E bounded, it holds from Proposition 6.3 above, that,  α ≥ 0, P α (E) = 0 6= ∅. We thus put   D = inf α ≥ 0, P α (E) = 0 = sup α ≥ 0, P α (E) = ∞ . Consider next the quantity M (E) defined previously. We obtain M (E)α ≤ Pα (E), ∀α > 0.

page 106

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Packing Measure and Dimension

In particular, if α > D, we get lim M (E)α = 0.

→0

Hence, ∆(E) < α, ∀α > D ⇒ ∆(E) ≤ D. Let now 0 < α < D. Then, P α (E) = +∞. Consequently, Pα (E) = +∞, for some , 0 <  < 1. As a result, there exists, (Bj )j , an -packing of E, such that, X |Bj |α ≥ 2. j

Let next for n ∈ N, An be the number of the Bj ’s, such that, 1 1 ≤ |Bj | < n . n+1 2 2 We have ∞ X X 1 An ( n )α . 2≤ |Bj |α ≤ 2 n=0 j So, whenever 0 < s < α < D, we get ∞ 1 X 1 n (1 − s ) ( ) = 1. 2 n=0 2s Consequently, for 1 α 1 1 ) ≥ ( s )n (1 − s ), 2n 2 2 we immediately observe that 1 1 (2α−s )n (1 − s ) ≤ An ≤ M n+1 (E). 2 2 Hence, An (

∆(E) ≥ α − s, ∀s > 0, which yields that ∆(E) ≥ α, ∀α < D. Consequently, ∆(E) ≥ D.

b4547-main

107

page 107

February 7, 2022 16:4

ws-book9x6

108

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Corollary 6.1. P α is not an outer measure on Rd . Proof. Let (En )n be a collection of subsets of Rd , such that, bounded. Let α > supn ∆(En ). We get

S

n

En is

P α (En ) = 0, ∀n. So, whenever P α is an outer measure, there holds that [ P α ( En ) = 0. n

Consequently, ∆(

[

En ) ≥ α.

However, ∆(

[ n

En ) = sup ∆(En ) < α, n n

which is contradictory. To get an outer measure, we put for E ⊂ Rd , ( ) X [ α α Pˆ (E) = inf P (En ); E ⊂ En , n

where (En )n is any covering of E with subsets En , n ≥ 0. Proposition 6.5. (1) It holds that, ∀α ≥ 0, nX o [ Pˆ α (E) = inf P α (En ) : E = En , and En is bounded for all n . n

n

(2) Pˆ α (E) ≤ P α (E), ∀E ⊂ Rd , and ∀α ≥ 0. (3) For all subset E ⊂ Rd , the map α 7→ Pˆ α (E) satisfies Pˆ α (E) < +∞ =⇒ Pˆ β (E) = 0, ∀β > α. (4) Pˆ α (E) = 0, ∀α > d, and ∀E ⊂ Rd . (5) Pˆ α is an outer metric measure on Rd , ∀α ≥ 0. Proof. (1) It is obvious as E is a covering of itself. (2) Denote Peα (E) the right hand quantity. We obviously observe that Pbα (E) ≤ Peα (E).

page 108

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Packing Measure and Dimension

109

On the other hand, let (En )n be an arbitrary covering of E, and denote, for all n, Fn = En ∩ Cn , where Cn is a cube of side [−n, n] that intersects En . We get a covering of E by a sequence of bounded sets (Fn )n . Therefore, X X Peα (E) ≤ P α (Fn ) ≤ P α (En ). n

n

Consequently, Peα (E) ≤ Pbα (E). bα

(3) As P (E) < ∞, there exists a covering (Ei )i of E such that P α (Ei ) < ∞, for all i. Consequently, for β > α, we get from Proposition 6.4, P β (Ei ) = 0, for all i. As a result, P β (Ei ) = 0. (4) Let α > d, and (En )n be a covering of E by bounded sets. It follows from Proposition 6.3, that P α (En ) = 0, for all n. Consequently, due to the fact that X Pbα (E) ≤ P α (Fn ), n

we get Pbα (E) = 0. (5) The inequality Pbα (∅) = 0 is trivial. Let A ⊆ B two subsets in Rn . We get X X Pbα (A) = inf P α (Ei ) ≤ inf P α (Ei ) = Pbα (B). [ [ A⊆ Ei i B⊆ Ei i i

i

Let now ε > 0, and (An )n be a countable collection of subsets in Rd . There exists (Ein )i,n such that X ε Pbα (An ) ≤ P α (Ein ) ≤ Pbα (An ) + n . 2 i It results that [ X X Pbα ( An ) ≤ P α (Ein ) ≤ Pbα (An ) + ε. n

bα

We now prove that P

n

i,n

is metric. Let A ⊆ B. We know that P α (A) ≤ P α (B).

Let (Ei )i be a covering of A ∪ B. We have  X α α α α b b P (A) + P (B) ≤ P (Ei ∩ A) + P (Ei ∩ A) i X ≤ P α (Ei ∩ (A ∪ B)) i

≤

X i

P α (Ei ).

.

page 109

February 7, 2022 16:4

ws-book9x6

110

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

This yields that Pbα (A) + Pbα (B) ≤ Pbα (A ∪ B). The equality results from the sub-additivity of Pbα . Definition 6.2. Pˆ α is said to be the packing measure of dimension α on Rd . 6.3

Packing dimension

Let E ⊂ Rd be bounded. As for the cases of Hα (E), and P α (E), there exists, for Pˆ α (E) also, a cutting-off value D ∈ R, satisfying bα (E) = +∞, for α < D , and P bα (E) = 0, for α > D. P Definition 6.3. The critical value D is known as the packing dimension of E, denoted DimE. Figure 6.1 illustrates the concept of the packing dimension graphically.

Fig. 6.1: The packing dimension of a set E.

Proposition 6.6. (1) (2) (3) (4)

0 ≤ DimE ≤ d, ∀E ⊂ Rd . Dim{a} = 0, ∀a ∈ Rd . E ⊂ F ⇒ DimE ≤ DimF , ∀E, F ⊂ Rd . S Dim En = supDimEn , ∀ (En )n a sequence of subsets of Rd . n

n

page 110

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Packing Measure and Dimension

b4547-main

111

(5) ∀E ⊂ Rd , E is Countable ⇒ DimE = 0. (6) ∀E ⊂ Rd , m(E) > 0, ⇒ DimE = d, where m stands for the Lebesgue measure on Rd . (7) ∀E ⊂ Rd , E is bounded ⇒ DimE ≤ ∆(E). (8) ∀E ⊂ Rd , 0 < Pˆ α (E) < ∞ ⇒ α = DimE. Proof. (1) DimE ≥ 0, by definition. From Proposition 6.5, we obtain, for all α > d, Pbα (E) = 0. Consequently, DimE ≤ α, ∀ α > d, which yields that DimE ≤ d. (2) We observe easily that for all α > 0, Pbα ({a}) = 0. Consequently, Dim{a} = 0. (3) For E ⊂ F , we have Pbα (E) ≤ Pbα (F ). Consequently, for α > DimF , we get Pbα (E) = 0. DimE ≤ α, ∀ α > DimF, which yields that DimE ≤ DimF . (4) From the previous assertion, we get sup DimEn ≤ Dim

[

n

En .

n

Conversely, for all α > sup DimEn , we have n

Pbα (

[

En ) ≤ n Pbα (En ) = 0.

n

Consequently, α ≥ Dim

[

En , ∀ α > sup DimEn . n

n

As a result, Dim

[

En ≤ sup DimEn . n

n

(5) Write E = {an , n ∈ N}. We get from previously, DimE = sup Dim{an } = 0. n

(6) Let (En )n be a covering of E. We have, X 0 < m(E) ≤ m(En ). n

page 111

February 7, 2022 16:4

ws-book9x6

112

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Therefore, there exists n0 ∈ N, such that, m(En0 ) > 0. Consequently, for all ε-packing (B(xi , ri ))i of En0 , we have X X X 0 < m(En0 ) ≤ m(B(xi , ri )) ≤ C (2ri )d ≤ C(2ε)d−α (2ri )α . i

i

i

α

As a consequence, P (En0 ) = ∞. As a result, X P α (En ) = ∞, i

for all coverings (En )n of E. This yields that Pbα (E) = ∞, for all α < d. Consequently, DimE ≥ d. By assertion 1, we get the equality. (7) For α > ∆(E), we get P α (E) = 0, which yields that Pbα (E) = 0. As a result, DimE ≤ ∆(E). (8) P α (E) > 0 yields that α ≤ DimE. Similarly, P α (E) < ∞ yields that α ≥ DimE. Consequently, α = DimE. Proposition 6.7. ∀ E ⊂ Rd , we have n o [ DimE = inf sup∆(En ), E = En , En bounded, ∀n . n

n

ˆ Proof. Let ∆(E) be the right hand term. We have ∆(En ) ≥ DimEn , ∀n. As a consequence, sup ∆(En ) ≥ supDimEn = DimE. n

n

Hence, ˆ ∆(E) ≥ DimE. Let next, α > DimE, then, Pˆ α (E) = 0. Therefore, there exists bounded subsets En , n ∈ N, such that, [ E = En , n

and X

P α (En ) < ∞.

n

As a result, ∆(En ) ≤ α, ∀n.

page 112

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Packing Measure and Dimension

b4547-main

113

Thus, ˆ ∆(E) ≤ α, ∀α > DimE, which yields that ˆ ∆(E) ≤ DimE.

Corollary 6.2. ∀ E ⊂ Rd , we have dimE ≤ DimE. Proof. Write E =

S

En . We have,

n

dimEn ≤ ∆(En ). Hence, ∀ n, we get dimE = sup dimEn ≤ sup ∆(En ) = DimE. n

6.4

n

Exercises for Chapter 6

Exercise 1. Let E1 ⊂ Rd1 , E2 ⊂ Rd2 . Show that dimE1 × E2 ≤ dimE1 + DimE2 , and that DimE1 × E2 ≤ DimE1 + DimE2 . Exercise 2. Let E1 ⊂ Rd1 and E2 ⊂ Rd2 , where d1 , d2 ∈ N. Show that, if E2 is bounded, we have dimE1 × E2 ≤ dimE1 + ∆(E2 ), and ∆(E1 × E2 ) ≤ ∆(E1 ) + ∆(E2 ). Exercise 3. Let C be the triadic Cantor set. Show that log 2 DimC = . log 3

page 113

February 7, 2022 16:4

114

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Exercise 4. For k ∈ N, let {Xi1 ,...,ik : ij = 1 or 2}, be the set of 2k intervals of length 3−k that occur in the usual construction of the middle-third Cantor set E, at the level k, and nested in the usual way. (1) Show that, setting µ(Xi1 ,...,ik ) = 2−k , leads to a measure on E. (2) Show that the same is true by setting µ(Xi1 ,...,ik ) = (1/3)n1 (2/3)n2 , where n1 , and n2 are the number of occurrences of the digits 1, and 2 respectively in (i1 , . . . , ik ). Exercise 5. Let {F1 , . . . , Fm } be an iterated function system consisting of similarity transformations satisfying the open set condition, with attractor E of dimension s ∈ R. Show that Hs (Fi (E) ∩ Fj (E)) = 0, if i 6= j. Exercise 6. Find the packing, and the box dimensions of the set X = {(1/p, 1/q) : p, q ∈ Z+ } ⊂ R2 . Exercise 7. Let E be the middle-third Cantor set. Find estimates for Hs (E), and P s (E), where s = log 2/ log 3. Exercise 8. Let 0 < λ < 1/2, and F1 , F2 : R → R be given by 1 1 F1 (x) = λx, and F2 (x) = x + . 2 2 Describe the attractor of {F1 , F2 }, and find an expression for its Hausdorff, and packing dimensions. Exercise 9. (1) Show that the hypotheses of Theorem 3.2 imply that P s (E) ≤ P0s (E) < ∞, where P s , and P0s are the packing measure, and pre-measure, (See Section 2.1). (2) Assume that Prs (E) > a−s ,

page 114

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Packing Measure and Dimension

b4547-main

115

and show that, there exists disjoint Balls B1 , . . . , Bm , with X |Bi |t > a−t , for t > s. (3) Let gi : E → E ∩ Bi , satisfying a|Bi ||x − y| ≤ |gi (x) − gi (y)|. Show that dimH E > s. Exercise 10. Let E be the subset of the middle-third Cantor set consisting of those numbers in [0, 1], with base 3 expansion containing only the digits 0, and 2, and where two consecutive digits 2 are not allowed. Let E1 = E ∩ [0, 1/3], and E2 = E ∩ [2/3, 0]. Show that dimH E1 = dimH E2 = log((1 +

√

5)/2)/ log 3.

page 115

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Chapter 7

Multifractal Analysis of Gibbs Type Measures

In this chapter, we investigate one of the most important concepts in fractal analysis; the so-called multifractal formalism for measures. This is a main common point with the physical meaning, and interactions with fractals, especially with the concept of Gibbs measures. Indeed, the word multifractal has been firstly introduced in [Frisch and Parisi (1985)], in the framework of statistical physics, by Frisch, and Parisi, when studying turbulence. To reach such a formalism, the multifractal analysis focuses on the singularities of the irregular object, such as the measure, and computes the so-called spectrum of singularities by means of the Hausdorff dimension of some sets associated to the measure, and known as the singularities sets. A simple way to describe the geometry, or the structure of the support of a measure µ is to compute its box dimension, i.e, the number d, such that, Nδ (E) has the same order as δ −d , where Nδ (E) is the optimal number of balls with diameter δ necessary to cover the support. For a measure µ, the multifractal formalism is reduced to the statistical characterization of the local scaling invariance of the measure. One covers the support of the measure µ by subsets E(α) = {x ∈ Rm ; µ(B(x, r)) ∼ rα }. The spectrum of singularities will be f α) = dim E(α). Based on some analogy with the statistical physics, the spectrum is evaluated by means of an auxiliary function τ (q) which has the same rule as the thermodynamical potential, such as the free energy. Let ! X log sup µ(Ui )q i

τ (q) =

− log r The multifractal formalism affirms that f (α) = inf (αq + τ (q)) . q

116

.

page 116

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Multifractal Analysis of Gibbs Type Measures

b4547-main

117

The sup above is taken over all the countable partitions of Support(µ) composed with coverings Ui ’s with diameters at most r. q is the analogous of the temperature inverse in thermodynamics. In some cases of statistical, and/or thermodynamical physics, the transfer of energy is governed by cascade models, in the sense that, a disturbance on a scale receives energy from a larger one, and transfers it to smaller scale disturbances. This defines a Gibbs measure. Under the assumption that the energy transfer rate is constant both in space, and in the cascade steps, we obtain the scaling law for the velocity |v(t + h) − v(t)|q ∼ |h|−q/3 . However, experimentally, there are some intermittency due to possible presence of strong fluctuations of the energy transfer, and dissipation. This breaks down the low cited above for large q. So one shall take into account the non-cascading or the non-self-similar transfer. In the present chapter, we focus on the study of the multifractal analysis, and the multifractal formalism for the class of Gibbs measures, and show that in this case, the validity of the multifractal formalism is proved naturally. We propose in this section to recall some facts about Gibbs measures, and to explain the idea of computing the spectrum of singularity in some situation, where the construction of such measures is possible. This is mainly based on the possibility of constructing Gibbs measures supported by the singularity set. The main results exposed here are based on [Brown et al (1992)]. However, more formulations, and cases may be found in the whole list [Barnsley (2000); Batakis and Heurteaux (2002); Beardon (1965); B´elair (1987); Ben Mabrouk (2005, 2007, 2008a,b,c,e); Ben Mabrouk and Ben Abdallah (2006); Ben Mabrouk and Aouidi (2011, 2012); Ben Mabrouk, Aouidi and Ben Slimane (2014, 2016); Ben Nasr (1994, 1997); Ben Nasr and Bhouri (1997); Ben Nasr, Bhouri and Heurteaux (2002); Billingsley (1960, 1961, 1965, 1968); Buck (1970, 1973); Choquet (1961); Collet et al (1987); David and Semmes (1997); Edgar (1998, 2008); Eggleston (1949); Falconer (1994, 1985, 1990); Frostman (1935); Frame and Cohen (2015); Frisch and Parisi (1985); Gervais (2009); Gurevich and Tempelman (1999); Heurteaux (1998); Hutchinson (1981); King (1995); Lambert (2012); Lucas (2000); Mandelbrot (1982, 1995, 1997, 1999, 2004); Menceur, Ben Mabrouk and Betina (2016); Menceur and Ben Mabrouk (2019); Mignot (1998); Ngai (1997); Nguyen (2001); Olivier (1998); Olsen (1995); Pesin (1997); Pesin and Climenhaga (2009); Peyriere (1992); Rand (1989); Riedi (1995); Robert (2020); Rogers (1970); Sierpinski (1916); Spear (1992); Triebel (1992); Veerman (1998); Wu (1998); Yeh (2014)].

page 117

February 7, 2022 16:4

118

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

In the sequel, and for the sake of simplicity, we will restrict to the Euclidean space R. The general case may be deduced in a natural way. Consider the unit interval [0, 1[, a sequence of partitions Fn , n ∈ N, and S denote F = n Fn . We also assume, for all n ∈ N, that Fn+1 is a refinement of Fn , and that ∀t ∈ [0, 1[, we have |In (t)| −→ 0 as n → +∞, where In (t) is the unique element of F that contains t. We consider the analogues of the Hausdorff measure Hα , the packing pre-measure P α , and the packing measure Bbα , introduced previously, by restricting the coverings, and the packings on elements of the collection F. We thus obtain, in a similar way, associated dimensions, which will be denoted here also dim, Dim, and ∆. Finally, for a real-valued function ϕ, we denote ϕL its Legendre transform defined, for α ∈ R, by ϕL (α) = inf (α(x + 1) − ϕ(x)). x∈R

7.1

The multifractal formalism

The idea to be exposed here is based on [Brown et al (1992)], where the authors constructed a type of partition function, easy to compute, and thus permits to evaluate the spectrum of singularities for measures. For n ∈ N, x, y ∈ R, and µ a Borel probability measure on [0, 1[, denote X 1 µ(I)x+1 |I|−y , Cn (x, y) = log n I∈Fn P where is taken on the elements I ∈ Fn , such that, µ(I) 6= 0. Denote next C(x, y) = lim sup Cn (x, y) ∈ R. n

It is straightforward that C is convex, increasing as a function of y, and decreasing relatively to the variable x. Denote next n o Ω = (x, y); C(x, y) < 0 . Lemma 7.1. The following assertions hold. (1) There exists a function ϕ : R → R, concave, nondecreasing, and satisfying, n o ˚ Ω = (x, y) ∈ R2 ; y < ϕ(x − 0) , where ˚ Ω stands for the topological interior of Ω. (2) If ϕ is finite on an open interval containing −1, then, it is also finite on [a, +∞[, for all a ∈ [−1, ∞[.

page 118

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Multifractal Analysis of Gibbs Type Measures

119

Proof. (1) Denote, for x ∈ R, ϕ(x) = sup{y ∈ R; C(x, y) < 0}. Let also, x1 < x2 be two real numbers. We immediately observe that C(x2 , y) < C(x1 , y); ∀ y ∈ R. As a consequence, C(x2 , y) < 0, ∀ y < ϕ(x1 ). Hence, y < ϕ(x2 ), ∀ y < ϕ(x1 ). By taking the sup on y, we get ϕ(x1 ) ≤ ϕ(x2 ). Let now (x1 , y1 ), and (x2 , y2 ) be, such that, y1 < ϕ(x1 ) and y2 < ϕ(x2 ). For all t ∈ [0, 1], we have, from the convexity of C, C(tx1 + (1 − t)x2 , ty1 + (1 − t)y2 ) ≤ tC(x1 , y1 ) + (1 − t)C(x2 , y2 ) < 0. This yields that ty1 + (1 − t)y2 < ϕ(tx1 + (1 − t)x2 ). Taking the sup on y1 , and y2 , we obtain tϕ(x1 ) + (1 − t)ϕ(x2 ) ≤ ϕ(tx1 + (1 − t)x2 ). Now, denote n o e = (x, y) ∈ R2 ; y < ϕ(x − 0) . Ω e It is easy to see that Ω e is open. If there exists We shall prove that ˚ Ω = Ω. an auxiliary open set O, such that, e ( O ( Ω, Ω we immediately deduce that, there exists a point (x, y), such that, e (x, y) ∈ O, and (x, y) ∈ / Ω. This point satisfies easily C(x, y) < 0 (⇐⇒ y < ϕ(x)), and y ≥ ϕ(x) (⇐⇒ C(x, y) ≥ 0). This is a contradiction. (2) This follows immediately from the concavity of ϕ.

page 119

February 7, 2022 16:4

ws-book9x6

120

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Denote now n o log µ(In (t)) Bα = t ∈ [0, 1[; lim sup ≤α , n→+∞ log |In (t)| and o n log µ(In (t)) ≥α . Vα = t ∈ [0, 1[; lim sup n→+∞ log |In (t)| The following result is the analogue of the one proved in [Brown et al (1992)]. Theorem 7.1. (1) For all α ≤ ϕ0 (−1 − 0), we have DimBα ≤ ϕL (α). (2) For all α ≥ ϕ0 (−1 + 0), we have DimVα ≤ ϕL (α), where ϕL is the Legendre transform of ϕ, defined above. Proof. For p ∈ N, and α, η > 0, denote   1 Bα (η, p) = t ∈ Bα ; ∀n, |In (t)| ≥ or µ(In (t)) ≥ |In (t)|η . p Let t ∈ Bα . There exists q ∈ N, such that, ∀n ≥ q, µ(In (t)) ≥ |In (t)|η . Let p be, such that,  

q [

 

1 < inf |I|, I ∈ Fj .   p j=1 It holds, for t ∈ Bα (η, p), with |In (t)| < p1 , that n > p. Thus, µ(In (t)) ≥ |In (t)|η . Consequently, Bα =

[

βα (η, p).

n

It suffices then to prove the inequality for Bα (η, p). To do it, observe that, for {Ij } an -packing of Bα (η, p), and 0 <  < p1 , we have X X |Ij |δ = |Ij |δ |Ij |ηt |Ij |−ηt , t > 0, j

≤ ≤

j X j X j

|Ij |δ−ηt µ(Ij )t |Ij |t+1−1 |Ij |.

page 120

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Multifractal Analysis of Gibbs Type Measures

b4547-main

121

As a result, the last right hand member is bounded if ηt − δ < ϕ(t − 1), which is equivalent to δ > ηt − ϕ(t − 1) > 0, which is always possible because Bα 6= ∅. Hence, ∆Bα (η, p) ≤ ηt − ϕ(t − 1), ∀t > 0, ∀n > α. Consequently, DimBα (η, p) ≥ ηt − ϕ(t − 1). Therefore, DimBα ≤ ηt − ϕ(t − 1), ∀t > 0, ∀n > α. Therefore, ∀η > α, we get DimBα ≤ inf t>−1 (ηt − ϕ(t − 1)) = inf(η(t + 1) − ϕ(t) = inf t≥−1 (η(t + 1) − ϕ(t)). So, finally, DimBα ≤ inf (α(t − 1) − ϕ(t)) = ϕ∗ (α). t≥−1

Denote for the next  Eα =

t ∈ [0, 1[, lim n

 log µ(In (t)) =α . log |In (t)|

The following result is due to [Brown et al (1992)]. Theorem 7.2. Assume that ϕ is differentiable at x, and that there exists a Borel probability measure µx on [0, 1[, such that, ∀ I ∈ F, 1 µ(I)x+1 |I|−ϕ(x) ≤ µx (I) ≤ M µ(I)x+1 |I|−ϕ(x) . M Then, dimEϕ0 (x) = ϕ∗ (ϕ0 (x)). The measure µx is called Gibbs measure.

page 121

February 7, 2022 16:4

ws-book9x6

122

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Proof. We show firstly that µx is supported on Eϕ0 (x) . Next, we apply Billingsley Theorem to µx . Indeed, denote n o log µ(In (t)) Vα∗ = t ∈ [0, 1[, lim inf ≥α . n→∞ log |In (t)| We will show that µx (Vϕ∗0 (x) ) = µx (Bϕ0 (x) ) = 1. Observe that \ Vα∗ Bα = Eα . So, for α < ϕ0 (x), we get µx (Vα∗ ) = 1. Assume now that ϕ0 (x) > 0, and let α be, such that, ϕ(x + t) − ϕ(x) 0 |In (t)|α }. We have P µx (An ) = µx (I)|I|αt |I|−αt , (t > 0) I∈Fn ;µ(I)>|I|α P ≤K µ(I)x+1 |I|−ϕ(x) µ(I)t |I|−αt I∈Fn ;µ(I)>|I|α P ≤K µ(I)x+t+1 |I|−(ϕ(x)+αt) . I∈Fn

Let next t be, such that, ϕ(x) + αt < ϕ(x + t), or equivalently, α< which is possible, because

ϕ(x + t) − ϕ(x) , t ϕ(x + t) − ϕ(x) . t→0 t

α < ϕ0 (x) = lim

Theorem 7.3. Let x be, such that, ϕ0 (x) exists. The following assertions hold. (1) µx (I) ≤ M µ(I)x+1 |I|−ϕ(x) if µ(x) 6= 0. (2) µx (I) = 0 for µ(x) = 0. (3) dimEϕ0 (x) = ϕ∗ (ϕ0 (x)) = (x + 1)ϕ0 (x) − ϕ(x). Remark 7.1. (1) Theorem 7.3 is valid if ϕ is finite on the interval J without −1. (2) If ϕ is differentiable at x, and If (x + 1)ϕ0 (x) − ϕ(x) = 0, then, dimEϕ0 (x) = 0.

page 122

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Multifractal Analysis of Gibbs Type Measures

7.2

b4547-main

123

Existence of Gibbs measures

Let A be the set of words composed from {0, 1, 2, . . . , p − 1} as an alphabet, p ≥ 2 in N. For n ∈ N, let An be the set of elements of A whom length equals n. Let next µ be a measure on A, and F = {Ia ∈ An }n≥1 a partition of [0, 1]. Denote `(a) = |Ia |. Assume finally that µ, and ` satisfy respectively 1 µ(a)µ(b) ≤ µ(ab) ≤ M µ(a)µ(b), M and 1 l(a)l(b) ≤ l(ab) ≤ l(a)l(b), L for some constants L, M > 0 fixed. We have the following result. Proposition 7.1. Denote for x, y ∈ R, ∗ X 1 µ(a)x+1 l(a)−y , Cn (x, y) = log n a∈An

and C(x, y) = lim Cn (x, y). Then, C(x, y) is finite, ∀x, y ∈ R. n

Proof. Denote, for n ∈ N, Zn =

X

µ(a)l(a).

a∈An

Hence, Cn (0, −1) = Next, observe that Zn+m =

1 log Zn , ∀n ∈ N. n

X

µ(ab)l(ab), ∀n, m ∈ N.

ab,a∈Am ,b∈An

We immediately obtain 1 Zn Zm ≤ Zn+m ≤ M LZn Zm , ∀n, m ∈ N. ML This may be written as | log Zn+m − log Zn − log Zm | ≤ log M L, ∀n, m ∈ N. Denoting, for n ∈ N, Un = log Zn , we get |Um+n − Un − Um | ≤ K, ∀n, m ∈ N. This yields that for some ` ∈ R, we have Un K − l| ≤ , ∀n ∈ N. | n n Consequently, log M L |Cn − C| ≤ , ∀n ∈ N. n

page 123

February 7, 2022 16:4

124

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Denote for the next, l0 (a) = µ(a)x l(a)−y . It follows from Proposition 7.1 that, for µ(a) 6= 0, we obtain 1 0 l (a)l0 (b) ≤ l0 (ab) ≤ K1 l0 (a)l0 (b). K1 We have the following result. Proposition 7.2. Whenever   lim sup µ(a)l(a)e−nc = 0, n→+∞

a∈A

there exists a Borel probability measure γ on [0, 1[, and a constant K ≥ 1, such that, 1 µ(a)l(a)e−nc ≤ γ(a) ≤ Kµ(a)l(a)e−nc , ∀a ∈ An . K Denote for n ∈ N, A∗n = {a ∈ An , µ(a) 6= 0}. It is obvious that in the proposition, we may restrict to A∗n instead of An . Whenever l(a) is not defined for a ∈ An , such that, µ(a) = 0, the hypothesis a ∈ A∗n is necessary. In the other cases, γ(a) = 0, whenever µ(a) = 0, the proposition may be expressed in another way. Proposition 7.3. Assume that h i lim sup µ(a)l(a)e−nc = 0, ∀ a ∈ A. n→∞ a∈A∗ n

There exists a probability measure γ on [0, 1[, and a constant K ≥ 1, such that, 1 µ(a)l(a)e−nc ≤ γ(a) ≤ Kµ(a)l(a)e−nc , ∀a ∈ A∗n . K (2) ∀ a ∈ A, µ(a) = 0 =⇒ γ(a) = 0. (1)

Proof. Let, for n ∈ N, ln be the function defined on [0, 1[ by ln (t) = |In (t)|, where for t ∈ [0, 1[, In (t) is the interval of An containing t. For s ∈ R, denote ϕs =

∞ X n=1

ln e−ns .

page 124

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Multifractal Analysis of Gibbs Type Measures

125

We observe that Z

X

ln dµ =

µ(a)l(a) = Zn ,

a∈An

and Z ϕs dµ =

∞ X

Zn e−ns .

n=1

Denote next for s > c, Z ϕs dµ < ∞.

Z(s) =

We obtain a Borel probability measure Ps on [0, 1[, expressed, for any Borel set B ⊂ [0, 1[, as Z 1 ϕs dµ. Ps (B) = Z(s) B Let next a = t1 ...tn ∈ An be fixed, and denote for 1 ≤ j ≤ n, aj = t1 ...tj . We have, for µ(a) 6= 0, Z Z(s)Ps (a) = ϕs dµ Ia ∞ Z X lj dµ)e−js = ( j=1

Ia

n Z X ( = j=1

lj dµ)e−js +

Ia

j=n+1

Whenever j ≥ n + 1, we get Z X Z lj dµ = ( Ia

b∈Aj−n

Z ∞ X (

lj dµ) =

Iab

X

lj dµ)e−js .

Ia

l(ab)µ(ab).

b∈Aj−n

For 1 ≤ j ≤ n, we have similarly Z lj dµ = l(ab)µ(a). Ia

As a result, Z(s)Ps (a) = µ(a)

n X

l(aj )e−js +

j=1

∞ X X

l(ab)µ(ab)e−(n+j)s .

j=1 b∈Aj

Now, observe that 

X b∈Aj

 X l(ab) µ(ab)  l(a)µ(a). l(ab)µ(ab) =  l(a) µ(a) b∈Aj

page 125

February 7, 2022 16:4

ws-book9x6

126

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Denote Z(n,a) =

X l(ab) µ(ab) b∈Aj

and Za (s) =

l(a) µ(a)

∞ X

,

Zj e−js .

j=1

We obtain Z(s)Ps (a) = µ(a)

n X

l(aj )e−js + l(a)µ(a)e−ns Za (s).

j=1

On the other hand, K 2 ec−s K −2 ec−s ≤ Za (s) ≤ , ∀a, c−s 1−e 1 − ec−s and

K −2 ec−s K 2 ec−s ≤ Z(s) ≤ , ∀a. 1 − ec−s 1 − ec−s

Consequently, 1 log Z(n,a) −→ C as n −→ +∞. n We now emphasize the case where µ(a) = 0, and Ps (a) = 0. Indeed, Ps is considered as a probability measure on R. We have Ps ([0, 1]) = 1. Let (sk )k ⊂ {s > c} be, such that, sk −→ c. Hence, (Psk ) is weakly k→+∞

convergent to a probability measure γ on R. Consequently, ∀a, Psk (a) −→ γ(a). k→+∞

It remains to prove that γ(∂Ia ) = 0. To do it, write a = t1 ...tp . Let t ∈ ˚ Ia , and Ib , Ic be, such that, z ∈ Ib ∪ Ic ⊂ ˚ Ia . Denote finally I˜ = Ib ˚ ∪ Ic . We have ˜ ˜ ≤ lim Ps (]I[). γ(z) ≤ γ(]I[) k k→+∞

On the other hand, ˜ ≤ ps ([Ib ∪ Ic [) psk (]I[) k = psk (Ib ) + psk (Ic ) n µ(b) X l(bj )e−jsk + K1 µ(b)l(b)e−nsk ≤ Z(b) j=1 n µ(c) X + l(cj )e−jsk + K1 µ(c)l(c)e−nsk . Z(c) j=1 Consequently,   ˜ ≤ K1 µ(b)l(b)e−nc + µ(c)l(c)e−nc . γ(z) ≤ γ(]I[)

page 126

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Multifractal Analysis of Gibbs Type Measures

b4547-main

127

Remark 7.2. Denote, for n ≥ 1, n o Kn = sup l(a), a ∈ Fn , and assume that lim sup n→+∞

log Kn < 0. n

Then, ϕ is finite on an open interval J 3 −1. Theorem 7.4. Let x ∈ R be, such that, (x + 1)ϕ0 (x) − ϕ(x) 6= 0. Then, there exists a Borel probability measure µx on [0, 1[, satisfying (1) for all a, such that, µ(a) 6= 0, 1 µ(a)x+1 l(a)−ϕ(x) ≤ µx (a) ≤ Kµ(a)x+1 l(a)−ϕ(x) . K (2) µx (a) = 0, for µ(a) = 0. Proof. Denote as previously, for µ(a) 6= 0, l0 (a) = µ(a)x l(a)−ϕ(x) . It suffices to prove that i. C(x, ϕ(x)) = 0. ii. lim ( sup µ(a)l0 (a)) = 0. n→+∞ a∈An

We start by proving assertion (ii) Let x < −1, and J an interval, such that, ϕ is finite on J. Denote ρ= Whenever ρ =

ϕ(t) . t−1 we obtain ϕ(−1) 6= 0, and the same for ϕ(−1) < 0.

Now, for x > −1, and ρ = sup

Example 7.1. Take p = 2, A = {0, 1}. Let λ0 , λ1 > 0, and λ0 + λ1 = 1. Consider l(a) = |Ia1 ...an | = λ1 ...λan , and µ(Ia1 ...an ) =

1 . 2n

We claim that • µ is quasi-Bernoulli. • l is almost-multiplicative. Evaluate next C(x, y), and compute ϕ on R. Example 7.2. With the same notations as in Example 7.1, in the case of Cantor sets, we consider µ(Ri ) = λi ,

and

0 elsewhere,

where Ri is the interval corresponding to the multi-index i. Remark, in this case, that, for i = (i1 , i2 , ..., in ), and j = (j1 , j2 , ..., jl ) two multi-indices, one has µ(Rij ) = λij = λi λj = µ(Ri )µ(Rj ). In this case, µ is a Gibbs measure. Show that the multifractal formalism holds.

page 128

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Multifractal Analysis of Gibbs Type Measures

7.3

129

Exercises for Chapter 7

Exercise 1. Prove Proposition 7.2. Exercise 2. Let Ω be a compact subset of Rd , and let ν be a measure defined on subsets of Ω. For a given s > 0, let B(s) be an s-covering of Ω. Let νj be the measure of a box Bj ∈ B(s), and let pj = νj /ν(Ω) be the normalized probability of the box Bj . Discarding from B(s) any box for which pj = 0, we assume that pj > 0, for all j, and let B(s) be the number of such boxes. Denote B(s) X P next j = , and for q ∈ R, j=1

< pq−1 >≡

X

pqj .

j

For q 6= 1, the generalized dimension of order q is Dq ≡

log < pq−1 > 1 lim , q − 1 s→0 log s

assuming that the limit exists. For q = 1, the generalized dimension of order 1 is D1 ≡ lim Dq . q→1

1. Show that if the probability measure ν is uniform on Ω, then Dq is independent of q. 2. Prove that Dq (as a function of q) is continuous at q = 1. 3. For the Cantor set, prove that if p1 = 13 , then Dq = 1 for all q. Exercise 3. For the same notations as in Exercise 7.3, define X Zq (s) ≡ [pj (s)]q . Bj ∈B(s)

Let the box sizes u, and v be, such that, u > v. For q 6= 1, define Dq (u, v) ≡

1 Zq (u) log Zq (u) − log Zq (v) = log( ), (q − 1)(log u − log v) (q − 1)(log uv ) Zq (v)

and H(s) ≡ −

X Bj ∈B(s)

pj (s) log pj (s).

page 129

February 7, 2022 16:4

130

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Show that D1 (u, v) ≡

H(v) − H(u) . log( uv )

Exercise 4. For an integer J ≥ 2, let P (J) be the continuous minimization problem  PJ q   j=1 Pj ,   subject to, P  Jj=1 pj = 1,    pj ≥ 0, ∀ j. Show that for q > 1, the solution of P (J) is pj = the optimal objective function value is J 1−q .

1 J

for each j, and that

Exercise 5. Consider a sequence {Fn }n of finite partitions {Vi } of [0, 1[, constituted of right-open intervals, such that, Fn+1 is a refinement of Fn . For t ∈ [0, 1[, assume that |I˜n (t)| −→ 0, as n → +∞, where In (t) is the unique element of Fn that contains t. Denote finally F = ∪n Fn . Let ξ : F −→ R+ , and denote   X  H(ξ) = lim inf ξ(Vj ); {Vj } an ε − recovering of [0, 1[ . ε→0   j

The function ξ is said to be a Frostman function,   if H(ξ) > 0, and, if 0 is an adherence value of the sequence sup ξ(V ) . V ∈Fn

n

Show that if ξ is a Frostman function, there exist a probability measure ν on [0, 1[, a constant M > 0, and a number ε > 0, such that, ∀V ∈ F, |V | ≤ ε :

ν(V ) ≤ M ξ(V ).

Exercise 6. Consider a Cantor type set C obtained by an iterated function system (I, Si ), 1 = 1, 2, where I = 0, 1. Denote for n ∈ N, and a multi-index i = (i1 , ..., in ) ∈ {1, 2}n , Ii = Si1 ◦ ... ◦ Sin ◦ (I), the corresponding cell in the n-th iteration. Consider next two sequences (λni )n≥1 in ]0, 1[, such that, λni + λni = 1, ∀n ≥ 1.

page 130

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Multifractal Analysis of Gibbs Type Measures

131

For x, and y real numbers, define ( ) X [ |λi |x |Ii |−y ; C ⊂ Ii and |Ii | ≤ ε , K(x, y) = lim inf ε→0

i

i

Cn (x, y) =

1 log n

X

|λi |x |Ii |−y ,

|i|=n

and C(x, y) = lim sup Cn (x, y). n→+∞

Let, finally, ϕ(x) = sup{y; C(x, y) < 0}. (1) Show that the function ϕ is non-decreasing, and concave. (2) Show that C(x, ϕ(x)) = 0 for all x. For the next part, consider, for j ∈ Z, the set Bj = {i : 2−j ≤ |Ii | < 2.2−j }. For i = (i1 , . . . , in ), denote λi = λ1i1 . . . λnin , αmin = lim inf inf

log λi , log |Ii |

αmax = lim sup sup

log λi . log |Ii |

j→∞ i∈Bj

and j→∞

i∈Bj

Assume now that K(p, ϕ(p)) > 0, α ∈ [αmin , αmax ], ϕ is differentiable at p, and α = ϕ0 (p).
(3) Show that d(α) = αp − ϕ(p) = inf αx − ϕ(x) . x

Exercise 7. With the same notations as in Exercise 7.3, let Tn = {Ii ; |i| = n}). For a measure µ supported by K, and x, y real numbers, denote ( ) X [ x −y K µ (x, y) = lim inf µ(Ii ) |Ii | ; K ⊂ Ii and |Ii | ≤ ε , ε→0

i

i

and K µ (x, y) = lim sup ε→0

( X i

) µ(Ii )x |Ii |−y ; K ⊂

[ i

Ii and |Ii | ≤ ε .

page 131

February 7, 2022 16:4

ws-book9x6

132

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

(1) Show that the set {K µ = 0} is convex. (2) Show that the function K µ is decreasing on x, and non-decreasing on y. (3) Show that, if K µ (x, y) is finite, then K µ (x + t, y − u) = 0, for t, u > 0. (4) Show that the function ϕµ : R −→ R, defined by K µ (x, y) = 0},

ϕµ (x) = sup{y;

is well defined, concave, and non-decreasing. [ (5) Let F(R) be the set of functions µ defined on T = Tn , and p > 0. n

Show that the set Ap = {µ ∈ F(R) ; K µ (p, ϕµ (p)) > 0}, is a convex subset of F(R). Exercise 8. Let k ∈ N be fixed, and λi > 0, i = 1, 2, ..., k, such that,

k X

λi < 1. Denote

i=1

next F (x) = log

k X

 λxi .

i=1

(1) Show that F is real analytic, decreasing, and convex on R. (2) Find a condition on the λi ’s, i = 1, 2, ..., k for which F is strictly convex. (3) Describe the asymptotic behaviour of F at ±∞. Exercise 9. Let g : R+ → R+ be a continuous function, satisfying the following hypothesis, i. g is non-decreasing. ii. g(0) = 0. iii. There exist a constant C > 0, such that, g(2r) ≤ Cg(r), ∀ r ≥ 0. We denote, for ε > 0, and E ⊂ Rd , ( ) [ X Hgε (E) = inf g(diam(Ui )); Ui ⊂ Rd , diam(Ui ) < ε, E ⊂ Ui , i

i

and Hg (E) = lim Hgε (E), ε↓0

and put diam the diameter in the Euclidean distance sense.

page 132

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Multifractal Analysis of Gibbs Type Measures

b4547-main

133

1. Show that Hg is an outer metric measure on Rd . Let µ : B(Rd ) → R+ be a non-vanishing finite measure. Let A be a Borel set,, and K ∈]0, +∞[ be constant. 2. Assume that, for all x ∈ A, we have µ(B(x, r)) < K. lim sup g(r) r→0+ Show that µ(A) ≤ kHg (A). 3. Assume that, for all x ∈ A, we have µ(B(x, r)) lim sup > K. g(r) r→0+ Show that µ(A) ≥ C −2 kHg (A). 4. Assume that µ(Rd \ A) = 0, and Z Z µ(dx)µ(dy) < ∞. g(|x − y|) d d R R Show that Hg (A) = ∞. 5. Assume finally that µ(Rd \ A) = 0, and that, there exists α > 0, satisfying Z Z µ(dx)µ(dy) < ∞. |x − y|α d d R R Prove that dimH (A) ≥ α. Exercise 10. Let β : R → R be a convex function. For a Borel finite measure µ on R, r > 0, and q ∈ R, consider the q-th power moment sums of µ, given by X Mr (q) = µ(A)q , where the sum is over the r-cubes A, for which µ(A) > 0. Denote next log Mr (q) β(q) = lim inf , r→0 − log r and log Mr (q) β(q) = lim sup . − log r r→0 Denote next β L , and β L the Legendre transforms of β, and β, respectively. Show that β L (α) ≤ inf {β(q) + αq}, −∞0

which is also not an outer measure, as it lacks the property of sub-additivity. To get an outer measure, we put X q,t Pµq,t (E) = inf P µ (Ei ). E⊆∪i Ei

i

Finally, we take as a convention (which in fact may be proved), q,t q,t Hµ,δ (∅) = Pµ,δ (∅) = 0,

or equivalently, Hµq,t (∅) = Pµq,t (∅) = 0. Proposition 8.1. Hµq,t , and Pµq,t are metric outer measures on Rd . Proof. We will split the proof into the following steps. (1) (2) (3) (4)

Hµq,t Hµq,t Pµq,t Pµq,t

is an outer measure. is metric. is an outer measure. is metric.

(1) Recall that Hµq,t is an outer measure, if it satisfies the following three assertions, (1.i) Hµq,t (∅) = 0. (1.ii) Hµq,t is monotone, in the sense that, for any subsets E ⊆ F ⊂ Rd , we have Hµq,t (E) ≤ Hµq,t (F ). (1.iii) Hµq,t is sub-additive, in the sense that, for any sequence (An )n of subsets of Rd , we have [ X Hµq,t ( An ) ≤ Hµq,t (An ). n

n

page 136

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Extensions to Multifractal Cases

137

So, let us develop each point. q,t (1.i) This is true by the construction convention on Hµ , and Hµq,t . (1.ii) Let E ⊆ F be two subsets of Rd . We have q,t

q,t

Hµq,t (E) = sup Hµ (A) ≤ sup Hµ (A) = Hµq,t (F ). A⊆F

A⊆E

Consequently, Hµq,t (E) ≤ Hµq,t (F ). (1.iii) Consider firstly a sequence (En )n of subsets in Rd , such that, X Hµq,t (En ) < ∞. n

Let δ,  > 0, and (B(xni , rni ))i be a centred δ-covering of En , for which, we have X  q,t µ(B(xni , rni ))q (2rni )t ≤ Hµ,δ (En ) + n . 2 i [ The collection (B(xni , rni ))n,i is then a centred δ-covering of En . As a n

result, XX q,t [ (µ(B(xni , rni )))q (2rni )t Hµ,δ ( En ) ≤ n n i X 
q,t ≤ Hµ,δ (En ) + n 2 n X q,t 
≤ Hµ (En ) + n 2 n X ≤ Hµq,t (En ) + . n

By letting δ, and  going to 0, we obtain X q,t [ H µ ( En ) ≤ Hµq,t (En ). n

n

Let now (An )n be a covering of F by subsets of Rd . We have   q,t q,t [ Hµ (F ) = Hµ (An F ) X n ≤ Hµq,t (An ∩ F ) n

≤

X n

Hµq,t (An ).

page 137

February 7, 2022 16:4

ws-book9x6

138

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Taking the sup on F ⊆

[

An , we get

n

Hµq,t (

[

An ) ≤

n

X

Hµq,t (An ).

n

(2) We now prove that Hµq,t is a metric measure on Rd . Indeed, consider two subsets E1 , E2 of Rd , such that, d(E1 , E2 ) > 0, and Hµq,t (E1 ∪ E2 ) < ∞. Let also 0 < δ < d(E1 , E2 ),  > 0, F1 ⊆ E1 , F2 ⊆ E2 , and (B(xi , ri ))i be a centred δ-covering of F1 ∪ F2 , such that, X q,t q,t (µ(B(xi , ri )))q (2ri )t ≤ Hµ,δ (F1 ∪ F2 ) + . Hµ,δ (F1 ∪ F2 ) ≤ i

Consider next I = { i; B(xi , ri ) ∩ F1 6= ∅ }

and J = { i; B(xi , ri ) ∩ F2 6= ∅ }.

The countable sets (B(xi , ri ))i∈I , and (B(xi , ri ))i∈J are centred δ-coverings of F1 , and F2 , respectively. Hence, X X q,t q,t Hµ,δ (F1 ) + Hµ,δ (F2 ) ≤ (µ(B(xi , ri )))q (2ri )t + µ(B(xi , ri ))q (2ri )t i∈I i∈J X = (µ(B(xi , ri )))q (2ri )t , i q,t

≤ Hµ,δ (F1 ∪ F2 ) + . which leads to q,t

q,t

q,t

Hµ (F1 ) + Hµ (F2 ) ≤ Hµ (F1 ∪ F2 ) +  ≤ Hµq,t (E1 ∪ E2 ) + . By letting  ↓ 0, and taking next the sup on F1 ⊆ E1 , and F2 ⊆ E2 , we obtain Hµq,t (E1 ∪ E2 ) ≥ Hµq,t (E1 ) + Hµq,t (E2 ). The other inequality comes from the subadditivity of Hµq,t . (3) To prove that Pµq,t is an outer measure, we have to show the analogues of the three assertions (1.i), (1.ii), and (1.iii) above for Pµq,t instead of Hµq,t . We thus designate by (3.i), (3.ii), and (3.iii) the new analogues.

page 138

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Extensions to Multifractal Cases

139

(3.i) This is obvious from the construction of Pµq,t . (3.ii) Let E ⊆ F be two subsets of Rd . We have X q,t X q,t P µ (Ei ) ≤ inf P µ (Ei ) = Pµq,t (F ). Pµq,t (E) = inf [ [ F⊆ Ei i E⊆ Ei i i

i

(3.iii) Let now (An )n be a sequence of subsets of Rd ,  > 0, and (Eni )i be a covering of An , such that, X q,t  P µ (Eni ) ≤ Pµq,t (An ) + n . 2 i It result, for all  > 0, that X [ X X q,t P µ (Eni ) ≤ Pµq,t (An ) + , Pµq,t ( An ) ≤ n

n

n

i

which leads to Pµq,t (

[

An ) ≤

n

X

Pµq,t (An ).

n

(4) We now prove that Pµq,t is a metric measure on Rd . So, let A, and B be two subsets of Rd , such that, d(A, B) > 0. We shall show that Pµq,t (A ∪ B) = Pµq,t (A) + Pµq,t (B). Indeed, since Pµq,t is an outer measure, it suffices to prove that Pµq,t (A ∪ B) ≥ Pµq,t (A) + Pµq,t (B). Without loss of generality, we may assume that Pµq,t (A∪B) < ∞. For  > 0, let (Ei )i be a covering of A ∪ B, such that, X q,t P µ (Ei ) ≤ Pµq,t (A ∪ B) + . i

Consider Fi = A ∩ Ei , and Hi = B ∩ Ei . Then, the collections (Fi )i , and (Hi )i are two coverings of A, and B, respectively. Moreover, Fi ∩ Hj = ∅ for all i, and j. Besides X q,t q,t Pµq,t (A) + Pµq,t (B) ≤ (P µ (Fi ) + P µ (Hi )). i

Now, recall that d(A, B) > 0, which implies that q,t

q,t

q,t

P µ (A ∪ B) = P µ (A) + P µ (B). It follows that Pµq,t (A) + Pµq,t (B) ≤

X i

q,t

P µ (Ei ) ≤ Pµq,t (A ∪ B) + .

page 139

February 7, 2022 16:4

ws-book9x6

140

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

The following proposition, based on the well known Besicovitch covering q,t theorem, gives a comparison between the quantities Hµq,t , Pµq,t , and P µ . Proposition 8.2. There exists a constant C > 0, such that, for all subset E ⊂ Rd , and for all q, t ∈ R, we have q,t

Hµq,t ≤ CPµq,t ≤ CP µ . Proof. The right inequality holds for all constant C > 0. We shall thus prove the left one. Let F ⊆ Rd , δ > 0, and 1 V = {B(x, ); x ∈ F }. 2
Let also (Bij )j 1≤i≤ξ be the ξ families of V defined in Besicovitch’s covering Theorem, such that, (Bij )i,j is a centred δ-covering of F , and (Bij )j is a centred δ-packing of F , for all i ∈ N. We have q,t Hµ,δ (F )

≤

ξ X X i=1


q

t

µ(Bij ) (2rij ) ≤

j

ξ X

q,t

i=1

As a result, q,t

q,t

Hµ (F ) ≤ ξP µ (F ). Consequently, if E ⊆

[

Ei , we get

i

[ Hµq,t (E) = Hµq,t ( (Ei ∩ E)) X i ≤ Hµq,t (Ei ∩ E) i

≤

X

sup F ⊆Ei ∩E

i

≤ξ

X

≤ξ

X

i

q,t

Hµ (F )

sup F ⊆Ei ∩E

q,t

P µ,δ (F ) = ξP µ,δ (F ).

q,t

P µ (F )

q,t

P µ (Ei ),

i

which gives the inequality stated, with C = ξ. Proposition 8.3. Let E be a subset of Rd . (1) There exists a unique element α0 ∈ R, such that,  ∞ for t < α0 , Hµq,t (E) = 0 for t > α0 .

page 140

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Extensions to Multifractal Cases

b4547-main

141

(2) There exists a unique element β0 ∈ R, such that, ( ∞ for t < β0 , q,t Pµ (E) = 0 for t > β0 . (3) There exists a unique element γ0 ∈ R, such that,  ∞ for t < γ0 , q,t P µ (E) = 0 for t > γ0 . Proof. (1) We claim firstly that 0

∀ t ∈ R, Hµq,t (E) < ∞ =⇒ Hµq,t (E) = 0, ∀ t0 > t.

(8.1)

Indeed, let δ > 0, F ⊆ E, and (B(xi , ri ))i be a centred δ-covering of F . We have X 0 q,t0 Hµ,δ (F ) ≤ (µ(B(xi , ri )))q (2ri )t i X 0 ≤ δ t −t (µ(B(xi , ri )))q (2ri )t . i

Therefore, q,t0

q,t0

0

H µ,δ (F ) ≤ δ t −t H µ,δ (F ), which leads to q,t0

Hµ (F ) = 0,

∀ F ⊆ E.

Consequently, 0

Hµq,t (E) = 0. We thus put 0

α0 = inf{ t ∈ R Hµq,t (E) = 0 }. (2) Let t ∈ R be such that, Pµq,t (E) < ∞. There exists a sequence (Ei )i of subsets of Rd satisfying [ q,t E⊆ Ei , and P µ (Ei ) < ∞, ∀ i. i 0

Consider next, t > t, δ > 0, and (B(xni , rni ))n a centred δ-packing of Ei . We have X X 0 0 (µ(B(xni , rni)))q (2rni )t ≤ δ t −t (µ(B(xni , rni)))q (2rni )t . n

n

page 141

February 7, 2022 16:4

ws-book9x6

142

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Therefore, q,t0

0

q,t

P µ,δ (Ei ) ≤ δ t −t P µ,δ (Ei ), which leads to q,t0

P µ (Ei ) = 0, ∀ i. Consequently, 0

Pµq,t (E) = 0. We thus put, as previously, β0 (E) = inf{t ∈ R; Pµq,t (E) = 0}. q,t

(3) Let t ∈ R be such that, P µ (E) < ∞. For all centred δ-packing (B(xi , ri ))i of E, and all t0 > t, we have X X 0 0 (µ(B(xi , ri )))q (2ri )t ≤ δ t −t (µ(B(xi , ri )))q (2ri )t . i

i

Consequently, q,t0

P µ (E) = 0. We thus set γ0 = inf{t ∈ R; Pµq,t (E) = 0}.

Definition 8.1. The cut-off values α0 , β0 , and σ0 are called, respectively, the multifractal generalizations of the Hausdorff dimension, packing, and the logarithmic index of E. We denote them, respectively, dimqµ (E), Dimqµ (E), and ∆qµ (E). The following proposition resumes some usual, and elementary characteristics of these dimensions. Proposition 8.4. (1) dimqµ , and Dimqµ are monotone. i.e., for E ⊆ F , we have i)

dimqµ (E) ≤ dimqµ (F ),

and ii)

Dimqµ (E) ≤ Dimqµ (F ).

page 142

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Extensions to Multifractal Cases

b4547-main

143

(2) dimqµ , and Dimqµ are σ-stables. i.e [ i) dimqµ ( En ) = sup dimqµ (En ), n

n

and ii)

Dimqµ (

[

En ) = sup Dimqµ (En ). n

n

Proof. (1.i) The monotony of dimqµ . Let E ⊆ F be two subsets of Rd , such that, dimqµ (F ) < ∞. We have Hµq,t (F ) = 0,

∀ t > dimqµ (F ).

It results, from the monotony of Hµq,t , that Hµq,t (E) = 0,

∀ t > dimqµ (F ).

dimqµ (E) ≤ t,

∀ t > dimqµ (F ).

Hence,

Consequently, dimqµ (E) ≤ dimqµ (F ). (1.ii) The monotony of Dimqµ . Let E ⊆ F be two subsets of Rd , such that, Dimqµ (F ) < ∞. Hence, Pµq,t (F ) = 0,

∀ t > Dimqµ (F ).

As Pµq,t is an outer measure, we deduce that Pµq,t (E) = 0,

∀ t > Dimqµ (F ).

Thus, Dimqµ (E) ≤ t,

∀ t > Dimqµ (F ).

Therefore, Dimqµ (E) ≤ Dimqµ (F ). (2.i) The σ-stability of dimqµ . Let (En )n be a sequence of subsets Rd , such that, [ dimqµ ( En ) < ∞. n

We have, En ⊆

[ m

Em , ∀ n.

page 143

February 7, 2022 16:4

ws-book9x6

144

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

We deduce, from assertion 1.i. above, that, for all n, [ dimqµ (En ) ≤ dimqµ ( En ). n

Therefore, [ sup dimqµ (En ) ≤ dimqµ ( En ). n

Let now, t >

n

sup dimqµ (En ). n

We have

Hµq,t (En ) = 0,

∀ n.

Hµq,t ,

It follows, from the sub-additivity of that [ Hµq,t ( En ) = 0, ∀ t > sup dimqµ (En ). n

n

Consequently, dimqµ (

[

En ) ≤ t,

∀ t > sup dimqµ (En ). n

n

As a result, dimqµ (

[

En ) ≤ sup dimqµ (En ). n

n

(2.ii) The σ-stability of Observing that

Dimqµ .

Let (En )n be a sequence of subsets Rd .

En ⊆

[

Em , ∀ n,

m

we get [ sup Dimqµ (En ) ≤ Dimqµ ( En ). n

n

Next, assume that sup Dimqµ (En ) < ∞, and consider t > sup Dimqµ (En ). n

n

We obtain, Pµq,t (En ) = 0, ∀ n. It follows, from the sub-additivity of Pµq,t , that [ Pµq,t ( En ) = 0, ∀ t > sup Dimqµ (En ). n

n

Consequently, Dimqµ (

[

En ) ≤ t,

∀ t > sup Dimqµ (En ). n

n

Hence, [ Dimqµ ( En ) ≤ sup Dimqµ (En ). n

n

page 144

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Extensions to Multifractal Cases

145

In the rest of the chapter, we will denote bµ,E (q) = dimqµ (E), Bµ,E (q) = Dimqµ (E), and Λµ,E (q) = ∆qµ (E). In the case where E is the support of µ, we will simply denote bµ (q) = dimqµ (E), Bµ (q) = Dimqµ (E), and Λµ (q) = ∆qµ (E). Proposition 8.5. The following assertions hold. (1) (2) (3) (4) (5)

bµ,E , Bµ,E , and Λµ,E are decreasing. Bµ,E , and Λµ,E are convex. 0 ≤ bµ (q) ≤ Bµ (q) ≤ Λµ (q), ∀ q < 1. bµ (1) ≤ Bµ (1) ≤ Λµ (1) = 0. bµ (q) ≤ Bµ (q) ≤ Λµ (q) ≤ 0, ∀ q > 1.

Proof. (1) The monotony of bµ,E . Let p ≥ q be two real numbers, F ⊆ E, δ > 0, and (B(xi , ri ))i a centred δ-covering of F . For all t ∈ R, we have X X (µ(B(xi , ri )))p (2ri )t ≤ (µ(B(xi , ri )))q (2ri )t . i

i

Hence, p,t

q,t

Hµ,δ (F ) ≤ Hµ,δ (F ). By taking the limit as δ ↓ 0, we obtain p,t

q,t

Hµ (F ) ≤ Hµ (F ). This leads to Hµp,t (E) ≤ Hµq,t (E). Consequently, Hµp,t (E) = 0,

∀ t > bµ,E (q).

bµ,E (p) < t,

∀ t > bµ,E (q).

Therefore,

Hence, bµ,E (p) ≤ bµ,E (q). We shall now prove the monotony of Bµ,E . Let (Ei )i be a covering of E by a sequence of subsets of Rd . If holds, for p ≥ q, that X p,t X q,t P µ (Ei ) ≤ P µ (Ei ). i

i

page 145

February 7, 2022 16:4

ws-book9x6

146

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Therefore, Pµp,t (E) ≤ Pµq,t (E). It results that Pµp,t (E) = 0,

∀ t > Bµ,E (q).

Bµ,E (p) < t,

∀ t > Bµ,E (q).

Consequently,

Thus, Bµ,E (p) ≤ Bµ,E (q). We now prove the monotony of Λµ,E . Let p ≥ q be two real numbers, δ > 0, and (B(xi , ri ))i be a centred δ-packing of E. We have µ(B(xi , ri ))p (2ri )t ≤ µ(B(xi , ri ))q (2ri )t , ∀ t ∈ R. It results that p,t

q,t

P µ,δ (E) ≤ P µ,δ (E). By taking the limit on δ ↓ 0, we obtain p,t

q,t

P µ (E) ≤ P µ (E). It follows that, p,t

P µ (E) = 0,

∀ t > Λµ,E (q).

Λµ,E (p) < t,

∀ t > Λµ,E (q).

Consequently,

Hence, Λµ,E (p) ≤ Λµ,E (q). (2) The convexity of Bµ,E . Consider p, q ∈ R, α ∈ [0, 1],  > 0, t = Bµ,E (q), and s = Bµ,E (p). We have Pµq,t+ (E) = Pµp,s+ (E) = 0. Consequently, There exits two coverings (Hi )i , and (Ki )i of E, such that, X q,t+ P µ (Hi ) ≤ 1, i

and X i

p,s+

Pµ

(Ki ) ≤ 1.

page 146

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Extensions to Multifractal Cases

147

Consider, for each n ∈ N, the set [

En =

(Hi ∩ Kj ).

1≤i,j≤n

As Pµq,t is an outer measure, we obtain Pµαq+(1−α)p,αt+(1−α)s+ (En ) n X ≤ Pµαq+(1−α)p,αt+(1−α)s+ (Hi ∩ Kj ) ≤

≤

i,j=1 n X

αq+(1−α)p,αt+(1−α)s+

Pµ

i,j=1 X n

q,t+ P µ (Hi

(Hi ∩ Kj )

α  X 1−α n p,s+ ∩ Kj ) (Hi ∩ Kj ) Pµ i,j=1

i,j=1 α 1−α

≤n n

= n < ∞.

Consequently, Bµ,En (αq + (1 − α)p) ≤ αt + (1 − α)s + ,

∀  > 0.

As a result, Bµ,E (αq + (1 − α)p) ≤ αBµ,E (q) + (1 − α)Bµ,E (p). We now show the convexity of Λµ,E , let p, q be two real numbers, α ∈ [0, 1], s > Λµ,E (p), and t > Λµ,E (q). Let also δ > 0, and (Bi = B(xi , ri ))i be a centred δ-packing of E. From H¨ older’s inequality, we get X (µ(Bi ))αq+(1−α)p (2ri )αt+(1−α)s i

≤

X α X 1−α (µ(Bi ))q (2ri )t (µ(Bi ))p (2ri )s , i

i

which leads to αq+(1−α)p,αt+(1−α)s

P µ,δ

 (E) ≤

α  1−α q,t p,s P µ,δ (E) P µ,δ (E) .

By taking the limit as δ ↓ 0, we obtain  α  1−α αq+(1−α)p,αt+(1−α)s q,t p,s Pµ (E) ≤ P µ (E) P µ (E) .

page 147

February 7, 2022 16:4

ws-book9x6

148

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Consequently, αq+(1−α)p,αt+(1−α)s

Pµ

(E) = 0.

Now, as s and t are arbitrary chosen, with s > Λµ,E (p), and t > Λµ,E (q), we deduce that Λµ,E (αq + (1 − α)p) ≤ αΛµ,E (q) + (1 − α)Λµ,E (p). (3) It follows, from Proposition 8.2, that bµ (q) ≤ Bµ (q) ≤ Λµ (q), ∀ p ∈ R.

(8.2)

(4) From assertion (3), it suffices to show that bµ (1) ≥ 0, Indeed, if t < 0, 0 < δ < Support(µ), we get X

1 2,

Λµ (1) ≤ 0.
and B(xi , ri ) i is a centred δ-covering of and

µ(B(xi , ri ))(2ri )t ≥ 1,

i

which leads to 1,t

Hµ,δ (Support(µ)) ≥ 1,

∀ t < 0.

Consequently, bµ (1) ≥ t,

∀ t < 0 ⇒ bµ (1) ≥ 0.
Consider now t > 0, 0 < δ < and B(xi , ri ) i a centred δ-packing of Support(µ). We have X 1,t P µ,δ (Support(µ)) ≤ µ(B(xi , ri ))(2ri )t ≤ 1. 1 2,

i

Thus, 1,t

P µ,δ (Support(µ)) ≤ 1,

∀ t < 0.

Therefore, Λµ (1) ≤ t,

∀ t < 0,

which yields, finally, that Λµ (1) ≤ 0. (5) is similar to assertion (3). Remark 8.1. (a) (b) (c) (d)

bµ,E (0) = dim E. Bµ,E (0) = Dim E. Λµ,E (0) = ∆(E). Bµ,E , and Λµ,E are finite on [0, +∞[.

page 148

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Extensions to Multifractal Cases

8.2

149

Generalized Bouligand–Minkowski dimension

We consider a Borel probability measure µ on Rd . For a > 1, and E a subset of Support(µ), we write
 µ B(x, ar) 
, Ta (E) = lim sup sup x∈E µ B(x, r) r↓0 and for x ∈ Support(µ), we write Ta (x) = Ta ({x}). Lemma 8.1. The two following assertions are equivalent. (1) ∀α > 1, Ta (E) < ∞. (2) There exists a > 1, such that, Ta (E) < ∞. Proof. The implication 1) =⇒ 2) is obvious. We shall prove the opposite sense. Let a > 1 be, such that, Ta (E) < ∞, and b > 1. There exists n ∈ N, such that, b ≤ an . Thus,
 µ B(x, br) 
Tb (E) = lim sup sup x∈E µ B(x, r)
r↓0  µ B(x, an r) 
≤ lim sup sup x∈E µ B(x, r) r↓0
n  Y µ B(x, ai r) 
= lim sup sup i−1 r) x∈E i=1 µ B(x, a r↓0  n ≤ Ta (E) < ∞.

In the sequel we denote P0 (Rd , E) = {µ ∈ P (Rd ), such that, ∃ a > 1; ∀ x ∈ E, Ta (x) < ∞}, P1 (Rd , E) = {µ ∈ P (Rd ), such that, ∃ a > 1; Ta (E) < ∞}, P0 (Rd ) = P0 (Rd , Support(µ)), and P1 (Rd ) = P1 (Rd , Support(µ)). For δ > 0, and q ∈ R, we write ( ) X

q
q Sµ,δ (E) = sup µ B(xi , δ) ; B(xi , δ) i packing of E , i

page 149

February 7, 2022 16:4

ws-book9x6

150

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

q Tµ,δ (E) = inf

( X

)

q
µ B(xi , δ) ; B(xi , δ) i covering of E

,

i

q C µ (E)

= lim sup δ↓0

C qµ (E) q Lµ (E)

= lim inf δ↓0

= lim sup δ↓0


q log Sµ,δ (E) − log δ
q log Sµ,δ (E)

,

− log δ
q log Tµ,δ (E) − log δ

,

,

and Lqµ (E)

= lim inf δ↓0


q log Tµ,δ (E) − log δ

.

We have the following result. Theorem 8.1. (1) For all q ≤ 0, we have (i) bµ,E (q) ≤ Lqµ (E) = C qµ (E). q (ii) Lµ,E (q) = C µ (E) = Λµ,E (q). (2) For all q > 0, we have (i) Lqµ (E) ≤ C qµ (E). q (ii) Lµ,E (q) ≤ C µ (E) ≤ Λµ,E (q). (3) If µ ∈ P0 (Rd ), and q > 0, we have q

bµ,E (q) ≤ Lµ (E). (4) If µ ∈ P1 (Rd ), and q > 0, we have (i) Lqµ (E) = C qµ (E). q (ii) Lµ,E (q) = C µ (E) = Λµ,E (q). Proof. (1.i) We will prove, in the first step, that, for all q ≤ 0, Lqµ (E) = C qµ (E),

q

q

and Lµ (E) = C µ (E).

According to Besicovitch’s covering Theorem, there exists A > 0, such that, for all q ∈ R, q q Tµ,δ (E) ≤ ASµ,δ (E).

page 150

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Extensions to Multifractal Cases

b4547-main

151

Hence, ∀ q ∈ R, we get Lqµ (E) ≤ C qµ (E),

q

q

and Lµ (E) ≤ C µ (E).

This proves assertion (2.i) and the first inequality of assertion (2.ii).

Conversely, consider a centred δ-packing B(xi , δ) i of E. Let B(yi , 2δ ) be δ a centred 2δ -covering E.
For all i ∈ N, let ki be, such that, xi ∈ B(yki , 2 ). It holds that B(xi , δ) i is a centred δ-packing of E. Moreover, for i 6= j, we have ki 6= kj . So, as q ≤ 0, we get !q X
q X δ
q µ(B(xi , δ)) µ(B(xi , δ)) = µ(B(yki , )) µ(B(yki , δ/2)) 2 i i X δ
q ≤ µ(B(yi , )) . 2 i Consequently, q Sµ,δ (E) ≤ Tµ,q δ (E). 2

Thus, ∀ q ≤ 0, it holds that Lqµ (E) ≥ C qµ (E),

q

q

and Lµ (E) ≥ C µ (E).

Now, assume that the right hand quantities in assertion (1.i) are finite. (The inequality is evident if these members are infinite). Let t > Lqµ (E), and F ⊆ E. There exists a sequence (δn )n ⊆]0, 1[, decreasing to 0, such that, q log(Tµ,δ (E)) n t> , ∀ n ∈ N. − log δn
As a result, ∀ n ∈ N, there exist a centred δn -covering B(xni , δn ) i of E, such that, X
q µ(B(xni , δn )) < δn−t . i

Without loss of generality, we may assume that all the balls meet the set F
. For each i fixed, choose an element yi ∈ B(xni , δn )∩F . Hence, B(yi , 2δn ) i is a centred δn -covering of F . Therefore, X
q q,t µ(B(xni , δn )) (4δn )t Hµ,2δn (F ) ≤ i !q X µ(B(yi , 2δn ))
q t =4 µ(B(xni , δn )) δnt µ(B(xni , δn )) i X
q ≤ 4t µ(B(xni , δn )) δnt i

≤

4t δn−t δnt

= 4t ,

page 151

February 7, 2022 16:4

ws-book9x6

152

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

which gives q,t

∀ F ⊆ E, and t > Lqµ (E).

Hµ (F ) ≤ 4t , Consequently,

∀ t > Lqµ (E).

Hµq,t (E) ≤ 4t < ∞, Finally, we get bµ,E (q) ≤ t,

∀ t > Lqµ (E),

which leads to bµ,E (q) ≤ Lqµ (E). (1.ii) It remains to prove the last equality. Let’s show that q

C µ (E) ≤ Λµ,E (q). The inequality is satisfied if the right member is infinite. Suppose, so, that Λµ,E (q) < +∞. Let t = Λµ,E (q),  > 0, and 0 < δ < 1 be, such that, q,t+

P µ,δ (E) < 1, ∀ 0 < δ < δ .
For a centred δ-packing B(xi , δ) i of E. we get X X
q
q µ(B(xi , δ)) = (2δ)−(t+) µ(B(xi , δ)) (2δ)t+ i

i q,t+

≤ (2δ)−(t+) P µ,δ (E) ≤ (2δ)−(t+) . Consequently, q Sµ,δ (E) ≤ (2δ)−(t+) ,

which gives that q

C µ (E) ≤ Λµ,E (q), ∀ q ∈ R. Remark that this gives also the last inequality in assertion (2.ii). We shall next show that q

Λµ,E (q) ≤ C µ (E). Suppose, as usually, that the right member is finite. Let t = Λµ,E (q),  > 0, and 0 < δ0 < 1. From the definition of Λµ,E (q), we have q,t+/2

∞ = Pµ

q,t−/2

(E) ≤ P µ,δ0

(E).

page 152

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Extensions to Multifractal Cases

153


There exists a centred δ0 -packing B(xi , ri ) i of E, such that, X
q 1< µ(B(xi , δ)) (2ri )t−/2 . i

For n ∈ N, Let In = { i ∈ N;

δ 2n+1

≤ ri
Lµ (E).

As the sequence (Em )m increases to E, we obtain bµ,E (q) ≤ t,

q

∀ t > Lµ (E),

which yields that q

bµ,E (q) ≤ Lµ (E). (4.i) It remains to show that Lqµ (E) ≥ C qµ (E)

q

q

and Lµ (E) ≥ C µ (E) ≥ Λµ,E (q).

So, recall that µ ∈ P1 (Rd , E). Consequently, there exists β > 0, and r0 > 0, such that, µ(B(x, 3r)) ≤ β, ∀ 0 < r < r0 and ∀ x ∈ E. µ(B(x, r))

page 154

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Extensions to Multifractal Cases

b4547-main

155

Let next 0 < δ < r0 , (B(xi , δ))i be a δ-packing of E, and (B(yi , 2δ ))i a centred 2δ -covering of E. For i ∈ N, let ki ∈ N be, such that, xi ∈ B(yki , 2δ ). It follows that !q  X X µ(B(xi , δ)) δ q )) µ(B(xi , δ)) = µ(B(y , k i 2 µ(B(yki , 2δ )) i i !q X µ(B(yki , 3δ ))  δ q 2 µ(B(y , )) ≤ ki 2 µ(B(yki , 2δ )) i !q X δ µ(B(yki , )) , ≤ βq 2 i which leads to q Sµ,δ (E) ≤ β q Tµ,q δ (E). 2

Consequently, q

q

Lqµ (E) ≥ C qµ (E) and Lµ (E) ≥ C µ (E). We shall now prove that q C µ (E) ≥ Λµ,E (q). Indeed, µ being in P1 (Rd , E), there exists consequently β > 0, and 0 < r0 < 1, satisfying µ(B(x, 2r)) ≤ β; ∀ 0 < r < r0 and ∀ x ∈ E. µ(B(x, r)) Let nest t = Λµ,E (q),  > 0, and 0 < δ0 < r0 . It follows that q,t−/2

Pµ (E) = ∞. As a result, there exists a δ0 -packing (B(xi , ri ))i of E, for which, we have q X 1< µ(B(xi , ri )) (2ri )t−/2 . i

Consider next IN , µN , and δ defined as in the proof of assertion (1.ii). We have q X q Sµ,δ (E) ≥ µ(B(xi , δ)) i∈IN !q q X µ(B(xi , δ0 /2N +1 ))  −q ≥β µ(B(x , r )) i i µ(B(xi , δ0 /2N )) i∈IN q X ≥ β −q µ(B(xi , ri )) i∈IN

≥ β −q µN > β −q C −1 (

δ0 −(t−) ) . 2N

page 155

February 7, 2022 16:4

ws-book9x6

156

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Consequently, q

C µ (E) ≥ Λµ,E (q).

In the sequel, for q ∈ R∗, and d > 0, we denote Z  q 1 q Iµ,δ = log µ(B(x, δ)) dµ(x), q Support(µ) q Iµ

= lim sup δ↓0

and I qµ = lim inf δ↓0

q Iµ,δ

− log δ q Iµ,δ

− log δ

,

.

WE have the following result. Proposition 8.6. The following assertions hold. (1) ∀ q ∈ R∗ , µ ∈ P1 (Rd ) we have q (i) C q+1 µ (Support(µ)) = qI µ . q+1

q

(ii) C µ (Support(µ)) = qI µ . (2) ∀ q < 0 we have (i) qI qµ ≤ Lq+1 µ (Support(µ)). q

q+1

(ii) qI µ ≤ Lµ (Support(µ)). (3) ∀ q > 0 we have q (i) C q+1 µ (Support(µ)) ≤ qI µ . q+1

q

(ii) C µ (Support(µ)) ≤ qI µ . Proof. (1) Assume that q < 0. As µ ∈ P1 (Rd ), there exists A > 0, and r0 > 0, for which we have µ(B(x, 2r)) < A ; ∀ x ∈ Support(µ), and 0 < r < r0 . µ(B(x, r)) Let 0 < δ < r0 , and (B(xi , δ))i be a δ-packing of Support(µ). We have q+1 X q Z X µ(B(xi , δ)) = µ(B(xi , δ)) dµ(x) B(xi ,δ) i i Z  q X ≤ A−q µ(B(x, 2δ)) dµ(x) i

≤ A−q

B(xi ,δ)

Z Support(µ)



µ(B(x, 2δ))

q

dµ(x).

page 156

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Extensions to Multifractal Cases

157

Therefore, q+1 q Sµ,δ (Support(µ)) ≤ A−q exp(qIµ,2δ ).

As a result, q+1

q

q C q+1 and C µ (Support(µ)) ≤ qI µ . µ (Support(µ)) ≤ qI µ   Conversely, for δ > 0, let B(xi , δ) be a centred δ-covering of Support(µ). i   Let also B(xij , δ) be the ξ families defined in the Besicovitch’s 1≤i≤ξ,j

covering Theorem. We have q+1 X q Z X µ(B(xij , δ)) = µ(B(xij , δ)) dµ(x) B(xij ,δ) i,j i,j Z  q X ≥ µ(B(x, 2δ)) dµ(x) i,j

B(xij ,δ)

Z



≥

q µ(B(x, 2δ)) dµ(x).

Support(µ)

Consequently, q+1 q ξSµ,δ (Support(µ)) ≥ exp(qIµ,2δ ).

As a result, we obtain q+1

q C q+1 µ (Support(µ)) ≥ qI µ

q

and C µ (Support(µ)) ≥ qI µ .   Assume now that q > 0. Let δ > 0, and B(xi , δ) be a δ-packing of i

Support(µ). We have q+1 X q Z X µ(B(xi , δ)) = µ(B(xi , δ)) dµ(x) B(xi ,δ) i i Z  q X ≤ µ(B(x, 2δ)) dµ(x) i

B(xi ,δ)

Z



≤

µ(B(x, 2δ))

q

dµ(x).

Support(µ)

It results that q+1 q Sµ,δ (Support(µ)) ≤ exp(qIµ,2δ ).

As a consequence, q C q+1 µ (Support(µ)) ≤ qI µ

q+1

q

and C µ (Support(µ)) ≤ qI µ .

page 157

February 7, 2022 16:4

ws-book9x6

158

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Conversely, as µ ∈ P1 (Rd ), there exists A > 0, and r0 > 0, satisfying µ(B(x, 2r)) < A ; ∀ x ∈ Support(µ) and 0 < r < r0 . µ(B(x, r)) Let 0 <  δ < r0 , (B(xi , δ))i be a δ-covering of Support(µ), and  B(xij , δ) the ξ families defined in the Besicovitch’s covering 1≤i≤ξ,j

Theorem. We get q+1 X q Z X µ(B(xij , δ)) = µ(B(xij , δ)) dµ(x) B(xij ,δ) i,j i,j Z  q X ≥ A−q µ(B(x, 2δ)) dµ(x) i,j

≥ A−q

B(xij ,δ)

Z



µ(B(x, 2δ))

q

dµ(x).

Support(µ)

As a consequence, q+1 q ξSµ,δ (Support(µ)) ≥ A−q exp(qIµ,2δ ).

Therefore, q+1

q C q+1 µ (Support(µ)) ≥ qI µ

q

and C µ (Support(µ)) ≥ qI µ .

Remark that we have also proved assertion 3).   (2) Let q < 0, δ > 0, and B(xi , δ) be a δ-covering of Support(µ). We i have q+1 X q Z X µ(B(xi , δ)) = µ(B(xi , δ)) dµ(x) B(xi ,δ) i i Z  q X ≥ µ(B(x, 2δ)) dµ(x) i

B(xi ,δ)

Z



≥

µ(B(x, 2δ))

q

dµ(x).

Support(µ)

It follows that q+1 q Tµ,δ (Support(µ)) ≥ exp(qIµ,2δ ).

As a result, q Lq+1 µ (Support(µ)) ≥ qI µ

q+1

q

and Lµ (Support(µ)) ≥ qI µ .

page 158

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Extensions to Multifractal Cases

8.3

159

The multifractal spectrum

This section is concerned with the computation of the spectrum of singularities for measures, in the framework of the generalized multifractal analysis. In this context, the possibility of being Gibbs measures may fail, compared to Chapter 7. Let µ be a Borel probability measure on Rd , x ∈ Rd , and put αµ (x) = lim inf

log(µ(B(x, r))) , log r

αµ (x) = lim sup

log(µ(B(x, r))) . log r

r↓0

and r↓0

These quantities define, respectively, the local lower dimension, and the local upper dimension of µ at the point x. In the case of equality, the common value is called the local dimension of µ in x, and will be denoted αµ (x). In the sequel, we consider, for all α ∈ R+ , the sets X α = {x ∈ Support(µ); αµ (x) ≥ α}, called the lower α-singularity set of the measure µ. Consider also α

X = {x ∈ Support(µ); αµ (x) ≤ α}, called, similarly, the upper α-singularity set of the measure µ. Their intersection α

X(α) = X α ∩ X , defines finally the α-singularity set of the measure µ. We will show that these sets are in the heart of the computation of the spectrum of singularities of µ. Indeed, we have the following definition. Definition 8.2. We call multifractal of a measure µ the function d : α 7→ d(α) = dim X(α), where dim is the Hausdorff dimension. In this chapter, we will study the link between this spectrum, and the functions bµ , Bµ , and Λµ introduced previously. Proposition 8.7. Consider α ≥ 0, and q ∈ R. The following assertions hold.

page 159

February 7, 2022 16:4

ws-book9x6

160

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

(1) If αq + bµ (q) ≥ 0, we have α

(i) dimX ≤ αq + bµ (q) (ii) dimX α ≤ αq + bµ (q)

for q ≥ 0 for q ≤ 0

(2) If αq + Bµ (q) ≥ 0, we have α

(i) DimX ≤ αq + Bµ (q) (ii) DimX α ≤ αq + Bµ (q)

for q ≥ 0 for q ≤ 0

Proof. (1.i) It suffices to prove that ( ∀ δ > 0, t ∈ R, and q ≥ 0, αq + t ≥ 0, we have (∗) Hαq+t+δ (X α ) ≤ 2αq+δ Hq,t (X α ). µ Indeed, if (∗) is true, we get α

Hαq+t+δ (X ) = 0,

∀ t > bµ (q), δ > 0.

Hence, α

dimX ≤ αq + t + δ,

∀ t > bµ (q), δ > 0,

which leads to α

dimX ≤ αq + bµ (q). We will now prove the property (∗). It is clear that this property is true for all q = 0. Assume so, that q < 0. For m ∈ N∗ , consider the set α

α

Xm = { x ∈ X ; Let 0 < η
0, t ∈ R, and q ≤ 0, αq + t ≥ 0, we have  Hαq+t+δ (X ) ≤ 2αq+δ Hq,t (X ), µ α α which in turn leads to dimX α ≤ αq + t + δ,

∀ t > bµ (q), δ > 0.

So finally, dimX α ≤ αq + bµ (q). (2.i) Let’s start by establishing the property ( ∀ δ > 0, t ∈ R, and q ≥ 0, αq + t ≥ 0, we have (∗∗) α α P αq+t+δ (X ) ≤ 2αq+δ Pµq,t (X ). Such property is obvious for q = 0. Suppose then that q > 0. For m ∈ N, 1 α α Consider the set X m defined in assertion 1), and Let E ⊆ X m , 0 < η < , m
and B(xi , ri ) i be a η-packing of E. We have (µ(B(xi , ri )))q (2ri )t ≥ 2t riαq+t+δ . Therefore, X X q,t (2ri )αq+t+δ ≤ 2αq+δ (µ(B(xi , ri )))q (2ri )t ≤ 2αq+δ P µ,η (E). i

i

Consequently, αq+t+δ

Pη

q,t

(E) ≤ 2αq+δ P µ,η (E),

∀ η > 0,

which leads to P

αq+t+δ

q,t

(E) ≤ 2αq+δ P µ (E),

α

∀ E ⊆ X m.

page 161

February 7, 2022 16:4

ws-book9x6

162

Fractal Analysis: Basic Concepts and Applications

Fractal Analysis: Basic Concepts and Applications α

Therefore, if (Ei )i is any covering of X m , we obtain P

αq+t+δ

α (X m )

αq+t+δ

[

α (X m

!

=P ∩ Ei ) i  α  X = P αq+t+δ X m ∩ Ei i

X

≤

P

αq+t+δ



α

X m ∩ Ei



i αq+δ

≤2

X

≤ 2αq+δ

X

q,t

Pµ



α

X m ∩ Ei



i q,t

P µ (Ei ).

i

We deduce that α

α

α

α

P αq+t+δ (X m ) ≤ 2αq+δ Pµq,t (X m ). Consequently, P αq+t+δ (X ) ≤ 2αq+δ Pµq,t (X ). Therefore, α

P αq+t+δ (X ),

∀ t > Bµ , δ > 0.

It results that α

DimX ≤ αq + t + δ,

∀ t > Bµ , δ > 0,

and finally, α

DimX ≤ αq + Bµ . (2.ii) By an analogue process to the previous assertion, we get ( ∀ δ > 0, t ∈ R, and q ≤ 0, αq + t ≥ 0 we have P αq+t+δ (X α ) ≤ 2αq+δ Pµq,t (X α ), which gives that P αq+t+δ (X α ) = 0,

∀ t > Bµ (q), δ > 0.

As a result, we get DimX α ≤ αq + t + δ,

∀ t > Bµ (q), δ > 0,

and finally, DimX α ≤ αq + Bµ (q).

b4547-main

page 162

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Extensions to Multifractal Cases

b4547-main

163

Proposition 8.8. ∀ q ∈ R, such that, αq + bµ (q) < 0 or αq + Bµ (q) < 0, we have X(α) = ∅. Proof. It suffices to prove that whenever αq +bµ (q) < 0 or αq +Bµ (q) < 0, α we get X α = ∅, for q ≤ 0, and X = ∅ for q ≥ 0. Indeed, let q ≤ 0, and suppose that X α 6= ∅. There exists x ∈ Support(µ), satisfying αµ (x) ≥ α. Then, for all  > 0, there exists a sequence (rn )n decreasing to 0, such that, 1 0 < rn < , and µ(B(x, rn )) < rnα− . n Consequently,  q µ(B(x, rn )) (2rn )t > 2t , where t = −q(α − ), which leads to Hµq,t ({x}) > 2t . As a result, bµ (q) ≥ dimqµ ({x}) ≥ t, ∀  > 0. Letting  → 0, we obtain bµ (q) ≥ −αt which is contradictory. α Let now q ≥ 0, and suppose that X 6= ∅. There exists x ∈ Support(µ), such that, αµ (x) ≤ α. Then, for all  > 0, there exists a sequence (rn )n decreasing to 0, such that, 1 and µ(B(x, rn )) > rnα+ . 0 < rn < n Denote t = −q(α + ). It holds that µ(B(x, rn ))q (2rn )t > 2t . It results, as previously, that Hµq,t ({x}) > 2t , which yields that bµ (q) ≥ t = −q(α + ). By letting  → 0, we end up with a contradiction. The same result can be proved by considering Bµ instead of bµ . We are now able to introduce, and prove the fundamental result in this chapter, which deals with the multifractal formalism for measures due to [Olsen (1995)]. Such a result applies the well known Borel-Cantelli lemma, and the large deviation formalism. We thus recall the two results before stating the fundamental theorem. A general case due to the mixed multifractal formalism for measures has been developed in [Menceur, Ben Mabrouk and Betina (2016); Menceur and Ben Mabrouk (2019)], where a generalized large deviation formalism has been established, for the case of non necessary Gibbs measures, and for a mixed multifractal analysis. Lemma 8.2. Borel-Cantelli Lemma. Let (Ω, A, P) be a probability space. For all An ∈ X A, we have P(An ) < ∞ =⇒ P(limAn ) = 0. n≥1

page 163

February 7, 2022 16:4

ws-book9x6

164

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Theorem 8.2. Large deviations formalism. Let (Wn )n be a sequence of random variables on a probability space (Ω, A, P), and (an )n ⊂]0, +∞[ with lim an = ∞. Let, for n ∈ N, n→+∞

  1 log E(exp(tWn )) , an and assume that Cn (t) is finite for all n, and p, and that Cn (t) =

C(t) = lim Cn (t), n→+∞

exists, and finite for all t. Then, (1) The function C is convex. 0 0 (2) If for t ∈ R, C− (t) ≤ C+ (t) < α, then  ! 1 −an C(t) lim sup log e E exp(tWn )1{ Wn ≥α} < 0. an n→+∞ an X (3) If e−an is finite for all  > 0, then n

lim sup n→+∞

Wn 0 ≤ C+ (0), an

P a.s.

0 0 (t), (t) ≤ C+ (4) If for t ∈ R, α < C−

!  1 −an C(t) lim sup < 0. log e E exp(tWn )1{ Wn ≤α} an n→+∞ an X e−an is finite, for all  > 0, then (5) Whenever n 0 C− (0) ≤ lim sup n→+∞

Wn , an

P a.s.

Proof. (1) It is sufficient to prove that the function Cn is convex. Indeed, this function is twice differentiable, and Z Z  Z 2 2 tWn tWn tWn Wn e dP e dP − Wn e dP Ω Ω Ω . Cn ”(t) = Z 2 tWn e dP Ω

The Cauchy-Schwartz’s inequality applied to the functions 1 1 f = Wn exp( tWn ), and g = exp( tWn ) 2 2 permits to affirm that Cn is convex.

page 164

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Extensions to Multifractal Cases

165

(2) As C is convex, there exists δ > 0, such that, C(t) + αδ − C(t + δ) > 0. Therefore,    1 log e−an C(t) E exp(tWn )1{ Wn ≥α} an an   Z 1 = etWn dP log e−an C(t) n an {W an ≥α}   Z 1 = log e−an (C(t)+αδ) etWn +an αδ dP n an {W a ≥α}   Z n 1 −an (C(t)+αδ) (t+δ)Wn e dP ≤ log e n an {W an ≥α}   1 −an (C(t)+αδ) ≤ log e E(exp((t + δ)Wn )) an   1 −an (C(t)+αδ−Cn (t+δ)) = log e an = −(C(t) + αδ − Cn (t + δ)). By taking the upper limit we get the result. (3) For n, m ∈ N, consider the set Tn,m = {

Wn 1 0 ≥ C+ (0) + }. an m

0 Take next in assertion (2), t = 0, and α = C+ (0) +

lim sup n

1 m.

We obtain

1 log P(Tn,m ) < 0. an

Hence, there exists  > 0, and N ∈ N, such that, lim sup n

1 log P(Tn,m ) < −, ∀n ≥ N, an

which leads to P(Tn,m ) < e−an . Consequently, the series

X

P(Tn,m ) is convergent. Borel-Cantelli Lemma

n

yields that P(lim sup Tn,m ) = 0, ∀m. n

page 165

February 7, 2022 16:4

ws-book9x6

166

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

As a results,   [ Wn 0 > C+ (0) = P( lim sup Tn,m ) = 0. P lim sup an n n m Consequently, lim sup n

Wn 0 ≤ C+ (0). an

Theorem 8.3. Let µ be a Borel probability measure on Rd , q a fixed real number. Let tq ∈ R, rq , K q , K q ∈]0, ∞[, νq be a Borel probability measure on Support(µ), and ϕq : R+ → R a function. Let (rq,n )n ⊂]0, 1[ be, such that, X log rq,n+1  rq,n ↓ 0, → 1, and rq,n < ∞ ∀ > 0. log rq,n n Assume next that H1 ) ∀ x ∈ Support(µ), and r ∈]0, rq [, Kq ≤ 

νq (B(x, r)) q ≤ Kq. µ(B(x, r)) (2r)tq exp(ϕq (r))

H2 )

ϕq (r) = o(log r), as r →Z 0.   p 1 H3 ) Cn (p) = log µ(B(x, rq,n )) dνq (x) exists, − log rq,n Support(µ) and finite, for all p, n ∈ N. H4 ) C(p) = lim Cn (p) exists, and finite for all p ∈ R. n→+∞

Then, the following assertions hold. (i) dim(X −C+0 (0) ∩ X (

0 −C− (0)

)≥

0 0 0 (0)q + Bµ (q) ≥ −C− (0)q + bµ (q) , q ≤ 0 (0)q + Λµ (q) ≥ −C− −C− 0 0 0 −C+ (0)q + Λµ (q) ≥ −C+ (0)q + Bµ (q) ≥ −C+ (0)q + bµ (q) , q ≥ 0

(ii) If moreover C is differentiable at 0, then fµ (−C 0 (0)) = b∗µ (−C 0 (0)) = Bµ∗ (−C 0 (0)) = Λ∗µ (−C 0 (0)). Proof. The important point of the proof is to show that the measure νq is supported by the set X −C+0 (0) ∩ X

0 −C− (0)

. This follows in fact from the

page 166

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Extensions to Multifractal Cases

b4547-main

167

Large Deviation Theorem. Indeed, denote t = tq , K = K q , K = K q , ϕ = ϕq , ν = νq , and rn = rq,n . For x ∈ Support(µ), we assume that h i log µ(B(x, rn ) αµ (x, rn ) = lim inf , n log rn and h i log µ(B(x, rn ) αµ (x, rn ) = lim sup

log rn

n

.

(i) We shall show that bµ (q) = Bµ (q) = Λµ (q) = t. Due to Proposition 8.8, it suffices to prove that t ≤ bµ (q),

and

Λµ (q) ≤ t.

Without loss the generality, we may assume that rq < 1. Let  > 0, and 0 < δ < rq be, such that, ϕ(r) log r < , ∀ r; 0 < r < δ .   Consider B(xi , ri ) a δ-packing of Support(µ). Due to the hypothesis H1 , i we obtain  q  q 2− K µ(B(xi , ri )) (2ri )t+ ≤ K µ(B(xi , ri )) (2ri )t eϕ(ri )   ≤ ν B(xi , ri ) . It holds that X

µ(B(xi , ri ))

q

 2 X  ν B(xi , ri ) K i  2 [ = ν B(xi , ri ) K i 2 ≤ . K

(2ri )t+ ≤

i

Hence, q,t+

P µ,δ (Support(µ)) ≤

2 , K

∀ 0 < δ < δ .

Consequently, Λµ (q) ≤ t + ,

∀  > 0.

page 167

February 7, 2022 16:4

ws-book9x6

168

Fractal Analysis: Basic Concepts and Applications

b4547-main

page 168

Fractal Analysis: Basic Concepts and Applications

This yields, finally, that, Λµ (q) ≤ t.  Consider now that -covering B(xi , ri ) of Support(µ). The hypothesis i H1 yields that  q  q 2 K µ(B(xi , ri )) (2ri )t− ≥ K µ(B(xi , ri )) (2ri )t eϕ(ri )   ≥ ν B(xi , ri ) . 

As a result, X

µ(B(xi , ri ))

q

(2ri )t− ≥

i

 2− 2− X  . ν B(xi , ri ) ≥ K i K

Therefore, q,t−

Hµ,δ (Support(µ)) ≥

2− , K

∀ 0 < δ < δ ,

which leads to bµ (q) ≥ t − ,

∀  > 0.

Hence, bµ (q) ≥ t. Consider now the set  0 0 (0) . M = x ∈ Support(µ); −C+ (0) ≤ αµ (x, rn ) ≤ αµ (x, rn ) ≤ −C− We will prove that M = X −C+0 (0) ∩ X −C−0 (0) . So, let n ∈ N, and r > 0 be, such that, rn+1 ≤ r ≤ rn . We have       log µ(B(x, r)) log rn log µ(B(x, rn )) log rn+1 log µ(B(x, rn+1 )) ≤ ≤ . log rn+1 log rn log r log rn log rn+1 This gives that for all x ∈ Support(µ), We have αµ (x) = αµ (x, rn ),

and αµ (x) = αµ (x, rn ).

Then the equality M = X −C+0 (0) ∩ X −C−0 (0) . We will apply now the Large deviation Theorem. So let’s take Ω = Support(µ), A = B(Support(µ)),

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Extensions to Multifractal Cases

169

P = µ, Wn (x) = log(µ(B(x, rn ))), and an = − log rn . It holds that 0 0 (0) ν. p.s. −C+ (0) ≤ αµ (x, rn ) ≤ αµ (x, rn ) ≤ −C−

Consequently, the measure ν is supported by M . If x ∈ Support(µ), and 0 < r < rn , the hypothesis H1 yields that, for all x ∈ M , ( 0 −qC− (0) + t for q ≤ 0, αµ (x) ≥ 0 −qC+ (0) + t for q ≥ 0. As ν(M ) = 1, we conclude due to Billingsley’s Theorem, that ( 0 −qC− (0) + t for q ≤ 0, dim M ≥ 0 −qC+ (0) + t for q ≥ 0. (2) If C is differentiable at 0, assertion (1) leads to dim M ≥ −qC 0 (0) + t ≥ Λ∗µ (−C 0 (0)) ≥ Bµ∗ (−C 0 (0)) ≥ b∗µ (−C 0 (0)). On the other hand, ν(M ) = 1. The set M is then non empty. Due to the Proposition 8.8, we obtain −qC 0 (0) + t ≥ 0. Consequently, dim M ≤ −qC 0 (0) + t, ∀ q ∈ R. By taking the lower bound on q, we obtain dim M ≤ b∗µ (−C 0 (0)) ≤ Bµ∗ (−C 0 (0)) ≤ Λ∗µ (−C 0 (0)).

Corollary 8.1. Assume that the hypotheses H1 , H2 , H3 , and H4 of Theorem 8.3 hold, and moreover C is differentiable at 0, then fµ (−C 0 (0)) = b∗µ (−C 0 (0)) = Bµ∗ (−C 0 (0)) = Λ∗µ (−C 0 (0)). Proof. The hypothesis H1 is true for all q ∈ R. Therefore, for all q < 0, x ∈ Support(µ), and r, 0 < r < rq , we have 1≤

h µ(B(x, 2r)) iq ν(B(x, 2r)) K ≤ 2t exp(ϕ(2r) − ϕ(r)) . ν(B(x, r)) K µ(B(x, r))

page 169

February 7, 2022 16:4

ws-book9x6

170

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Therefore, for all x ∈ Support(µ), and r, 0 < r < rq µ(B(x, 2r)) < β, µ(B(x, r)) where  β=

K −t 2 K

1/q

1 exp( [ϕ(r) − ϕ(2r)]). q

Let (B(xij , rn ))1≤iξ,j be the ξ families defined in Besicovitch’s covering Theorem extracted from the family (B(xi , rn ))i . We will proceed by steps. Step 1: p + q − 1 ≥ 0. If p ≥ 0, we have Z h ip+q−1 µ(B(y, rn )) dµ(y) K(2rn )t exp(ϕ(rn )) B(xij ,rn )

≤ K(2rn )t exp(ϕ(rn ))

Z

h ip+q−1 µ(B(xij , 2rn )) dµ(y)

B(xij ,rn )

ip+q ≤ K(2rn )t exp(ϕ(rn )) µ(B(xij , 2rn )) h ip ≤ β p+q µ(B(xij , rn )) ν(B(xij , rn )) Z h ip ≤ β p+q µ(B(xij , rn )) dν(y) h

B(xij ,rn )

≤ β p+q

Z

h

ip µ(B(y, 2rn )) dν(y),

B(xij ,rn )

which leads to p+q−1

(p + q − 1)I µ

≤ Cq (p) + tq .

Conversely, Z

t

h ip+q−1 µ(B(y, 2rn )) dµ(y)

K(2rn ) exp(ϕ(rn )) B(xij ,rn )

≥ K(2rn )t exp(ϕ(rn ))

Z

h

µ(B(xij , rn ))

B(xij ,rn )

h ip+q ≥ K(2rn )t exp(ϕ(rn )) µ(B(xij , rn )) Z h ip ≥ β −p µ(B(y, rn )) dν(y). B(xij ,rn )

Consequently, (p + q − 1)I µp+q−1 ≥ Cq (p) + tq .

ip+q−1

dµ(y)

page 170

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Extensions to Multifractal Cases

171

We suppose now that p ≤ 0. We have

K(2rn )t exp(ϕ(rn ))

Z

h

µ(B(y, rn ))

ip+q−1

dµ(y)

B(xij ,rn )

Z

t

h ip+q−1 µ(B(xij , 2rn )) dµ(y)

≤ K(2rn ) exp(ϕ(rn )) B(xij ,rn )

ip+q ≤ K(2rn )t exp(ϕ(rn )) µ(B(xij , 2rn )) h ip ≤ β p+q µ(B(xij , rn )) ν(B(xij , rn )) Z h ip ≤ β p+q µ(B(xij , rn )) dν(y) h

B(xij ,rn )

≤β

p

Z

h ip µ(B(y, rn )) dν(y).

B(xij ,rn )

As a result, p+q−1

(p + q − 1)I µ

≤ Cq (p) + tq .

Conversely, Z

t

h ip+q−1 µ(B(y, 2rn )) dµ(y)

K(2rn ) exp(ϕ(rn )) B(xij ,rn )

Z

t

≥ K(2rn ) exp(ϕ(rn ))

h

µ(B(xij , rn ))

B(xij ,rn )

h ip+q ≥ K(2rn )t exp(ϕ(rn )) µ(B(xij , rn )) Z h ip −p ≥β µ(B(y, rn )) dν(y). B(xij ,rn )

Therefore, (p + q − 1)I µp+q−1 ≥ Cq (p) + tq . Step 2: p + q − 1 ≤ 0.

ip+q−1

dµ(y)

page 171

February 7, 2022 16:4

ws-book9x6

172

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

For p ≤ 0, We have

K(2rn )t exp(ϕ(rn ))

Z

h

µ(B(y, 2rn ))

ip+q−1

dµ(y)

B(xij ,rn )

Z

t

h

≤ K(2rn ) exp(ϕ(rn ))

µ(B(xij , rn ))

ip+q−1

dµ(y)

B(xij ,rn )

h ip+q ≤ K(2rn )t exp(ϕ(rn )) µ(B(xij , rn )) h ip ≤ µ(B(xij , rn )) ν(B(xij , rn )) Z h ip ≤ µ(B(xij , rn )) dν(y) B(xij ,rn )

≤ β −p

Z

h

ip µ(B(y, rn )) dν(y).

B(xij ,rn )

This yields that p+q−1

(p + q − 1)I µ

≤ Cq (p) + tq .

Conversely, Z

t

h ip+q−1 µ(B(y, rn )) dµ(y)

K(2rn ) exp(ϕ(rn )) B(xij ,rn ) t

≥ K(2rn ) exp(ϕ(rn ))β

p+q−1

Z

h

µ(B(xij , rn ))

B(xij ,rn )

h ip+q ≥ K(2rn )t exp(ϕ(rn ))β p+q−1 µ(B(xij , rn )) Z h ip p+q−1 ≥β µ(B(y, 2rn )) dν(y). B(xij ,rn )

Consequently, (p + q − 1)I µp+q−1 ≥ Cq (p) + tq .

ip+q−1

dµ(y)

page 172

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Extensions to Multifractal Cases

173

We suppose now that p ≥ 0. We have Z h ip+q−1 K(2rn )t exp(ϕ(rn )) µ(B(y, 2rn )) dµ(y) B(xij ,rn )

≤ K(2rn )t exp(ϕ(rn ))

Z

h

µ(B(xij , rn ))

ip+q−1

dµ(y)

B(xij ,rn )

h ip+q ≤ K(2rn )t exp(ϕ(rn )) µ(B(xij , rn )) h ip ≤ µ(B(xij , rn )) ν(B(xij , rn )) Z h ip ≤ µ(B(xij , rn )) dν(y) B(xij ,rn )

Z

h

≤

ip µ(B(y, 2rn )) dν(y).

B(xij ,rn )

As a result, p+q−1

(p + q − 1)I µ

≤ Cq (p) + tq .

Conversely, K(2rn )t exp(ϕ(rn ))

Z

h

ip+q−1 µ(B(y, rn )) dµ(y)

B(xij ,rn )

≥ K(2rn )t exp(ϕ(rn ))β p+q−1

Z

h

ip+q−1 µ(B(xij , 2rn )) dµ(y)

B(xij ,rn )

h ip+q ≥ K(2rn )t exp(ϕ(rn ))β p+q−1 µ(B(xij , rn )) Z h ip µ(B(y, rn )) dν(y). ≥ β q−1 B(xij ,rn )

Therefore, (p + q − 1)I µp+q−1 ≥ Cq (p) + tq . We deduce that for all p, q ∈ R, (p + q − 1)Iµp+q−1 = Cq (p) + tq . As a result, (p + q − 1)Iµp+q−1 = Cµp+q (Support(µ)) = Λµ (p + q). Consequently, Cq (p) = Λµ (p + q) − Λµ (p).

page 173

February 7, 2022 16:4

ws-book9x6

174

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

Thus, if Λµ is differentiable at q, Cq will be differentiable at 0, and Cq0 (0) = Λ0µ (q). The Large Deviation theorem implies that αµ (x) = −Cq0 (0) ; νq , for almost every x ∈ support(µ). Finally, αµ (x) = −Λ0µ (q).

Corollary 8.2. Assume that the hypotheses H1 , H2 , H3 , and H4 of Theorem 8.3 hold for all q ∈ R. The following assertions hold. (i) If Bµ is differentiable at q, then αµ = −Bµ , νq , a.s, and {−Bµ0 (q); Bµ0 exists } ⊆ αµ (support(µ)). (ii) fµ = Bµ∗ on { −Bµ0 (q); Bµ0 exists }. Proof. Let q, such that, Λ0µ (q) exists. So Cq0 (0) exists. Corollary 8.1 leads to fµ (−C 0 (0)) = Λ∗µ (−C 0 (0)).

8.4

Exercises for Chapter 8

Exercise 1. Develop a proof of Lemma 8.2 (the Borel-Cantelli Lemma). Exercise 2. Let Ω be the set of open bounded intervals in R, and ` be the set function written on the `(a, b) = (b − a)s , for some real number parameter s. Let next the set function X H(A) = inf { `i (Ik ); A ⊂ ∪k Ik , Ik ∈ Ω, ∀k}. k

(1) For what values of s the set function H is an outer measure on Ω? (2) Evaluate the behavior of `s (x − r, x + r), as r goes to 0.

page 174

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Extensions to Multifractal Cases

175

Exercise 3. Let h(r) = rα (log | log r|)β , where α, β are constants. Consider ( ) X [ h H (E) = lim inf h(B(xi , ri )); xi ∈ E ⊂ B(xi , ri ), ri < ε . ε&0

i

i

(1) Study the possible Hausdorff measure properties of H. (2) Study the same problem with packings instead of coverings. Exercise 4. Let h(r) = rα (log | log r|)β , where α, β are constants. For a Borel probability measure µ on R, Consider for q ∈ R the set function, )

( h,q Hµ (E)

= lim inf ε&0

X

q

µ(B(xi , ri )) h(B(xi , ri )); xi ∈ E ⊂

i

[

B(xi , ri ), ri < ε

i

1) Study the possible Hausdorff measure properties of Hµh,q . 2) Study the same problem with packings instead of coverings. Exercise 5. Consider the function f (t) = −

γ(log t)γ−1 −(| log t|)δ e X]0,1[ (t), t

where γ > 0 is a constant, and the measure µ(t) = f (t)dt. (1) Compute αµ (0). (2) Show that for x ∈ (0, 1), log µ(B(x, r)) = −(log x)2 −

2 log x r + rε(r), x

r → 0.

(3) Deduce αµ (x), for x ∈ [0, 1]. (4) Let ϕ(r) = log r(| log r|)γ−1 , and νq,t (B(x, r)) = (µ(B(x, r)))q etϕ(r) . Show that  q γ−1 νq,t (B(x, 2r)) µ(B(x, 2r)) ∼ eγ(log 2)| log r| , as r → 0. νq,t (B(x, r)) µ(B(x, r))

.

page 175

February 7, 2022 16:4

ws-book9x6

176

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

(5) Deduce that γ−1 νq,t (B(x, 2r)) ∼ Cq,µ eγ(log 2)| log r| → ∞ (or 0) as r → 0. νq,t (B(x, r))

Exercise 6. Let ϕ : R+ → R be, such that, ϕ is non-decreasing, and ϕ(r) < 0, for r small enough, and consider, for a metric space (X, d), the function hq,t (r) = µ(B(x, r))q etϕ(r) , r > 0, x ∈ X, q ∈ R. Consider the quantity q,t

Hµ,ϕ, (E) = inf{

X

hq,t (r) },

i

where the inf is taken over the set of all centered -coverings of E, and for q,t the empty set, Hµ, (∅) = 0. Denote next q,t

q,t

q,t

Hµ,ϕ (E) = lim Hµ,ϕ, (E) = sup Hµ,ϕ, (E), ↓0

δ>0

and finally, q,t

q,t Hµ,ϕ (E) = sup Hµ,ϕ (F ). F⊆E

is an outer metric measure on Rd , for which, Borel sets (1) Show that are measurable. q,t . (2) Develop the multifractal analysis of Hµ,ϕ q,t Hµ,ϕ

Exercise 7. Let ϕ : R+ → R be, such that, ϕ is non-decreasing, and ϕ(r) < 0, for r small enough, and consider for a metric space (X, d) the function hq,t (r) = µ(B(x, r))q etϕ(r) , r > 0, x ∈ X, q ∈ R. Consider the quantity q,t

P µ,ϕ, (E) = sup{

X (µ(B(xi , ri )))q etϕ(ri ) }, i

where the sup is taken over the set of all centered -packings of E. For the q,t empty set, we set, as usual, P µ,ϕ, (∅) = 0. Next, we consider the limit as  ↓ 0, q,t

q,t

q,t

P µ,ϕ (E) = lim P µ,ϕ, (E) = inf P µ,ϕ, (E), ↓0

>0

page 176

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Extensions to Multifractal Cases

b4547-main

177

and finally, q,t Pµ,ϕ (E) =

inf

E⊆ ∪i Ei

X

q,t

P µ,ϕ (Ei ).

i

q,t (1) Show that Pµ,ϕ is an outer metric measure on Rd , for which, Borel sets are measurable. q,t (2) Develop the multifractal analysis of Pµ,ϕ .

Exercise 8. Let ϕ : R+ → R be, such that, ϕ is non-decreasing, and ϕ(r) < 0, for r small enough. Let µ be a Borel probability measure satisfying the assumption ! µ(B(x, ar)) Aµ,ϕ (a, α) = lim sup sup eαϕ(r) < ∞, µ(B(x, r)) r→0 x∈Sµ for some a > 1, and for all α > 0, where Sµ is the support of µ. Assume further that q,Λµ,ϕ (q) Hµ,ϕ (Sµ ) > 0,

for some q ∈ Rk+ . Show that there exists a Borel probability measure ν supported by Sµ , such that,  q ν(B(x, r)) ≤ K µ(B(x, r)) eΛµ,ϕ (q)ϕ(r) ; ∀x ∈ Sµ , ∀0 < r 1, and for all α > 0, where Sµ is the support of µ. Assume further that q,Bµ,ϕ (q) Hµ,ϕ (Sµ ) > 0,

for some q ∈ Rk+ . Show that there exists a Borel probability measure ν supported by Sµ , such that,  q ν(B(x, r)) ≤ K µ(B(x, r)) eBµ,ϕ (q)ϕ(r) ; ∀x ∈ Sµ , ∀0 < r 0 for some q ∈ Rk− . Show that there exists a Borel probability measure ν supported by Sµ , such that,  q ν(B(x, r)) ≤ K µ(B(x, r)) eΛµ,ϕ (q)ϕ(r) ; ∀x ∈ Sµ , ∀0 < r 0 for some q ∈ Rk− . Show that there exists a Borel probability measure ν supported by Sµ , such that,  q ν(B(x, r)) ≤ K µ(B(x, r)) eBµ,ϕ (q)ϕ(r) ; ∀x ∈ Sµ , ∀0 < r 0, and n ∈ N be, such that, 2 1 ≤ 2r < n . n 7 7 There exists a multi-index i = (i1 , i2 , ..., in ) ∈ {1, 4, 7}n , such that, Ii is the unique interval of Gn , which intersects I(x, r). Such interval has surely a predecessor Ij of Gn+1 contained in I(x, r). (j ∈ {1, 4, 7}). It holds consequently that µ(Iij ) ≤ µ(I(x, r)) ≤ µ(Iij ), and ν(Iij ) ≤ ν(I(x, r)) ≤ ν(Iij ). Denote next    (min pi )t (max ri )β(t) (max pi )t (max ri )β(t) i i i i K = inf (min pi )t (min ri )β(t) , , , i  i  (min ri )β(t) (min pi )β(t) i

i

and K = sup

 

 

1 1 1 , , .  (min pi )t (min ri )β(t) (min pi )t (min ri )β(t)  i

i

i

We have νq (I(x, r)) iq ≤ K. K≤h µ(I(x, r)) (2r)t Using Theorem 8.3, we obtain Cq (t) = −t

log 3 . log 7

i

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Some Applications

b4547-main

199

It is a differentiable function, with, Cq0 (t) = −

log 3 . log 7

From Corollary 8.2, we obtain dim X( 9.8

log 3 log 3 )= . log 7 log 7

Exercises for Chapter 9

Exercise 1. Consider the Sierpinski triangle ST constructed from the equilateral trian√ 1 3 gle of vertices O(0, 0), A(1, 0), and B( , ) in the plane by subdividing it 2 2 into four smaller congruent equilateral triangles, and removing the central triangle, and repeating the same removal action with each of the remaining smaller triangles infinitely. (1) Find three similarities on R2 yielding ST as the associated unique nonempty compact invariant set. (2) Write an algorithm (code) to illustrate the process graphically. (3) Compute the Hausdorff dimension of ST . Exercise 2. Denote K the Koch snowflake constructed from the equilateral triangle √ 3 1 ) by dividing ABC in the plane, with vertices O(0, 0), A(1, 0), and B( , 2 2 each line segment into three segments of equal length to draw an equilateral triangle with the middle segment as its base, and points outward, and removing the line segments that are bases of the new triangles. The Koch snowflake K is the limit as the above steps are followed infinitely. (1) Find suitable similarities on R2 yielding K as the associated unique non-empty compact invariant set. (2) Write an algorithm (code) to illustrate the process graphically. (3) Compute the perimeter of K. (4) Compute the area of K. (5) Compute the Hausdorff dimension of K. (6) Compute the volume of the solid of revolution of K about the axis of 1 symmetry x = of the triangle OAB. 2 Exercise 3. Let f : [0, 1] → R be, such that, f (x) = 3xχ[0,1/2[ (x) + 3(1 − x)χ[1/2,1] (x).

page 199

February 7, 2022 16:4

200

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Fractal Analysis: Basic Concepts and Applications

For n ∈ N, denote f n = F ◦ f ◦ ... ◦ F , n times, and let Λn = {x ∈ [0, 1]; f k (x) ∈ [0, 1], for k = 0, 1, ..., n}, and Λ = {x ∈ [0, 1]; f n (x) ∈ [0, 1], for all n ≥ 0}. (1) (2) (3) (4) (5) (6) (8)

Draw the graphs of f , Λ1 , and Λ2 . Give the endpoints of the intervals in Λ1 , and Λ2 . Give the endpoints of the intervals in Λn . Is Λ a Cantor set? Justify your answer. Compute the Lebesgue measure of Λ. Compute the box counting dimension of Λ. Find an iterated function system whose limit is Λ.

Exercise 4. Consider in R2 the set X constructed as follows. We start by the square [0, 1] × [0, 1]. Next, in the first iteration, we subdivide the original square into nine equal squares, and remove the upper right two squares, and the bottom left two squares (in the same direction, horizontal or vertical both of them). In the second iteration, we subdivide each of the remaining squares into nine equal squares, and remove the upper right two squares, and the bottom left two squares by the same way as in iteration 1. We continue to infinity. (1) Compute the Lebesgue measure of X. (2) Compute the box counting dimension of X. (3) Find an iterated function system whose limit is X.

page 200

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Some Applications

b4547-main

201

Exercise 5. Consider the so-called fractal antenna FA, applied in cellular phones, and constructed as illustrated by a deterministic iterated function system program, with the five first steps of iteration as in Figure 9.15 below.

Fig. 9.15: The first five iterations of a fractal antenna. (1) Find the smallest number of transformations needed for this iterated function system. (2) Write the iterated function system rules. (3) Compute the Lebesgue measure of FA. (4) Compute the box counting dimension of FA.

page 201

B1948

Governing Asia

This page intentionally left blank

B1948_1-Aoki.indd 6

9/22/2014 4:24:57 PM

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Bibliography

Alimohammady, M., Cattani, C. and Kalleji, M. K. (2017) Invariance and existence analysis for semilinear hyperbolic equations with damping and conical singularity. J. Math. Anal. Appl., 455(1), 569–591. Anguera, J., Andujar, A., Jayasinghe, J., Chakravarthy, V. and Chowdary, P. S. R. (2020) Fractal Antennas: An Historical Perspective. Fractal and Fract., 4(1). Arfaoui, S., Rezgui, I. and Ben Mabrouk, A. (2017) Wavelet Analysis On The Sphere, Spheroidal Wavelets, Degryuter, (with Sabrine Arfaoui and Imen Rezgui). Degruyter, 2017, ISBN 978-3-11-048188-4. Aversa, V. and Bandt, C. (1990) The Multifractal Spectrum of Discrete Measures. Acta Univ. Carolinae-Math. Et. Phys., 31(2), 5–8. Azizieh, C. (2002) Mod´elisation de S´eries Financi`eres par un Mod`el Multifractal. M´emoire pour diplˆ ome d’actuaire. Universit´e Libre de Bruxelles. Badea, A. F., Platon, M. L., Crisan, M., Cattani, C., Badea, I. and Pierro, G. (2013) Fractal analysis of elastographic images for automatic detection of diffuse diseases of salivary glands: Preliminary results. Computational and Mathematical Methods in Medicine, 1–6. Baker, A. and Schmidt, W. M. (1970) Diophantine approximation and Hausdorff dimension. Proc. London Math. Soc., 21, 1–11. Baker, R. C. (1976) Dyadic Methods in the measure theory of numbers. Transactions of the AMS, 221, 419–432. Baleanu, D. (2012) Wavelet Transforms and Their Recent Applications in Biology and Geoscience. InTechOpen, 2012. Baleanu, D. (2012) Advances in Wavelet Theory and Their Applications in Engineering, Physics and Technology. InTechOpen, 2012. Baleanu, D. (2015) Wavelet Transform and Some of Its Real-World Applications. InTechOpen, 2015. Baleanu, D., Guvenc, Z. B. and Tenreiro Machado, J. A. (2010) New Trends in Nanotechnology and Fractional Calculus Applications. Springer. Baleanu, D., Machado, J. A. T., Cattani, C., Baleanu, M. C. and Yang, X. J. (2014) Local fractional variational iteration and decomposition methods

203

page 203

February 7, 2022 16:4

204

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Bibliography

for wave equation on Cantor sets within local fractional operators. Abstr. Appl. Anal., 82, 1–6. Barlow, M. T. and Taylor, S. J. (1992) Defining fractal subsets of Zd . Proc. London Math. Soc., 64(3), 125–152. Barnsley, M. F. (2000) Fractals Everywhere. Second edition, Morgan Kaufmann. Barnsley, M. F., Devaney, R. L., Mandelbrot, B. B., Peitgen, H.-O., Saupe, D. and Voss, R. F. (1988) The Science of Fractal Images, Heinz-Otto Peitgen. In: Dietmar Saupe (eds.), Springer-Verlag New York. Barral, J., Nasr, F. B. and Peyri`ere, J. (2003) Comparing multifractal formalism: The neighbouring box condition. Asian J. Math., 7, 149–66. Batakis, A. and Heurteaux, Y. (2002) On relations between entropy and hausdorff dimension of measures. Asian J. Math., 6(3), 399–408. Beardon, A. F. (1965) On the Hausdorff dimension of general Cantor sets. Proc. Camb. Phil. Soc., 61, 679–694. B´elair, J. (1987) Sur le calcul de la dimension fractale. Ann. SC. math. Qu´ebec, 11(1), 7–23. Benaych-Georges, F. (2009) Introduction au calcul stochastique pour la finance, Cours, Master 1, Universit´e Paris VI, 2009. Ben Mabrouk, A. (2005) Multifractal analysis of some non isotropic quasi-selfsimilar functions. Far East J. Dyn. Syst., 7(1), 23–63. Ben Mabrouk, A. (2006) Wavelet Analysis of Anisotropic Quasi-Self-Similar Functions in a Nonlinear Case, Colloque WavE 2006, Polytechnic Federal School of Lausanne, Switzerland, 10–14 July. Ben Mabrouk, A. (2007) Study of some nonlinear self-similar distributions. International Journal of Wavelets, Multiresolution and Information Processing, 5(6), 1–10. Ben Mabrouk, A. (2008) Higher order multifractal formalism. Stat. Proba. Lett., 78, 1412–1421. Ben Mabrouk, A. (2008) On some nonlinear non isotropic quasi self similar functions. Nonlinear Dyn., 51, 379–398. Ben Mabrouk, A. (2008) An adapted group dilation anisotropic multifractal formalism for functions. J. Nonlinear Math. Phys., 15(1), 1–23. Ben Mabrouk, A. (2008) Anouar Ben Mabrouk: Higher order multifractal measures. Stat. Probab. Lett., 78, 1412–1421. Ben Mabrouk, A. (2008) Wavelet analysis of nonlinear self-similar distributions with oscillating singularity. Int. J. Wavelets Multiresolution Inf. Process., 6(3), 1–11. Ben Mabrouk, A. and Ben Abdallah, N. (2006) Study of some multinomial cascades. Adv. Appl. Stat., 6(3), 95–303. Ben Mabrouk, A., Ben Abdallah, N. and Dhifaoui, Z. (2008) Wavelet decomposition and autoregressive model for the prevision of time series. Appl. Math. Comput., 199(1), 334–340. Ben Mabrouk, A., Ben Abdallah, N. and Hamrita, M. E. (2011) A wavelet method coupled with quasi self similar stochastic processes for time series approximation. Int. J. Wavelets, Multiresolution Inf. Process., 9(5), 685–711.

page 204

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Bibliography

b4547-main

205

Ben Mabrouk, A. and Aouidi, J. (2011) Multifractal analysis of weighted quasiself-similar functions. Int. J. Wavelets Multiresolution Inf. Process., 9(6), 965–987. Ben Mabrouk, A. and Aouidi, J. (2012) Lecture Note on Wavelet Multifractal Analysis of Self Similarities. Lampert Academic Publishing, Verlag. ISBN 978-3-8465-9646-3. Ben Mabrouk, A., Aouidi, J. and Ben Slimane, M. (2014) A wavelet multifractal formalism for simultaneous singularities of functions. Int. J. Wavelets, Multiresolution Inf. Process., 12(1). Ben Mabrouk, A., Aouidi, J. and Ben Slimane, M. (2016) Mixed multifractal analysis for functions: General upper bound and optimal results for vectors of self-similar or quasi-self-similar of functions and their superpositions. Fractals, 24(4), 1650039. Ben Nasr, F. (1994) Analyse multifractale de mesures, CRAS. Paris, 319(I), 807–810. Ben Nasr, F. (1997) On some regularity conditions of Borel measures on R. J. Austral. Math. soc. S´erie A, 63, 218–224. Ben Nasr, F. and Bhouri, I. (1997) Spectre multifractal de mesures bor´eli`ennes sur Rd . C. R. Acad. Sci. Paris, t. 325, S´erie I, 253–256. Ben Nasr, F., Bhouri, I. and Heurteaux, Y. (2002) The validity of the multifractal formalism: Results and examples. Adv. Math., 165, 264. Benzi, R., Paladin, G., Parisi, G. and Vulpiani, A. (1984) On the multifractal nature of fully developed turbulence and chaotic systems. J. Phys, A-17, 3521–3531. Berglund, N. (2014) Martingales et calcul stochastique, Master 2 Recherche de Math´eematiques, Universit´e d’Orl´eans. Berglund, N. and Gentz, B. (2006) Noise-Induced Phenomena in Slow-Fast Dynamical Systems A Sample-Paths Approach, Probability and Its Applications. Springer. Bernay, M. (1975) La dimension de Hausdorff de l’ensemble des nombres r-d´eterministes. C.R. Acad. Sc. Paris, S´erie A, 280, 539–541. Besicovitch, A. S. (1934) On the sum of digits of real numbers represented in the dyadic system. Math. Ann., 110, 321–330. Beyer, W. A. (1962) Hausdorff dimension of level sets of some redemacher series. Pacific J., 12, 35–46. Billingsley, P. (1960) Hausdorff dimension in probability theory. III. J. Math., 4, 187–209. Billingsley, P. (1961) Hausdorff dimension in probability theory. II. III. J. Math., 5, 291–298. Billingsley, P. (1965) Ergodic Theory and Information. Wiley, NewYork. Billingsley, P. (1968) Convergence of Probability Measures. Wiley, NewYork. Billingsley, P. and Henningsen, I. (1975) Hausdorff dimension of some continuedfraction sets. Z. Wahrscheinlichkeitstheorie verw. Geb., 31, 163–173. Biswas, A., Cresswell, H. P. and Bing, C. S. (2012) Application of Multifractal and Joint Multifractal Analysis in Examining Soil Spatial Variation: A Review. Chapter 6 in Fractal Analysis and Chaos in Geosciences. Edited by Sid-Ali Ouadfeul, InTechOpen, pp. 109–138. DOI.org/10.5772/51437.

page 205

February 7, 2022 16:4

206

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Bibliography

Boyd, D. W. (1973) The residual set dimension of the Apollonian packing. Mathematika, 20, 170–174. Bozkus, S. K., Kahyaoglu, H. and Lawali, A. M. M. (2020) Multifractal analysis of atmospheric carbon emissions and OECD industrial production index. International Journal of Climate Change Strategies and Management, 12(4), 411–430. DOI: 10.1108/IJCCSM-08-2019-0050. Briggs, J. P. (1992) Fractals: The Patterns of Chaos: Discovering a New Aesthetic of Art, Science, and Nature. Simon & Schuster. Brown, C., Witschey, W. R. T. and Liebovitch L. S. (2005) The broken past: Fractals in archaeology. J. Archaeol. Method Th., 12(1), 37–78. Brown, G., Michon, G. and Peyri`erre, J. (1992) On the multifractal analysis of measures. J. Stat. Phys., 66, 775–779. Buck, R. J. (1970) A generalized Hausdorff dimension for functions and sets. Pacific J., 33, 170–174. Buck, R. J. (1973) Hausdorff dimensions for compact sets in Rn . Pacific J., 44, 421–434. Budzinski, T. (2019) Processus al´eatoires, TD. ENS Paris, 2017–2019. Cabrelli, C. A., Hare, K. E. and Molter, U. M. (1997) Sums of cantor sets. Ergod. Theory Dyn. Syst., 17(6), 1299–1313. Cabrelli, C. A., Hare, K. E. and Molter, U. M. (2002) Sums of Cantor Sets yielding an interval. J. Aust. Math. Soc. Series A, 73(3), 405–418. Cabrelli, C. A., Mendevil, F., Molter, U. M. and Shonkwiler, R. (2004) On the Hausdorff h-Measure of Cantor Sets. Pac. J. Math., 217(1), 45–59. Cabrelli, C. A., Molter, U. M., Paulauskas, V. and Shonkwiler, R. (2005) The Hausdorff dimension of p-Cantor sets. Real Analysis Exchange, 30(2), 413–433. Cajar, H. (1981) Billingsley dimension in probability spaces. Lecture notes in mathematics, 892. New York: Springer. Calvet, L. and Fisher, A. (2008) Multifractal Volatility, Theory, Forecasting, and Pricing. Academic press advanced finance series, 1st edn., September. Castiglioni, P. and Faini, A. (2019) A fast DFA Algorithm for multifractal multiscale analysis of physiological time series. Front. Physiol., 10, 18, Article: 115. DOI: 10.3389/fphys.2019.00115. Cattani, C. and Karaca, Y. (2018) Computational Methods for Data Analysis. De Gruyter. Cattani, C. and Rushchitsky, J. (2007) Wavelet and wave analysis as applied to materials with micro or nanostructure. Series on Adv. Math. Appl. Sci., 74. Cattani, C. and Pierro, G. (2013) On the fractal geometry of DNA by the binary image analysis. Bull. Math. Biol., 75(9), 1544–1570. Cattani, C. and Ciancio, A. (2016) On the fractal distribution of primes and prime-indexed primes by the binary image analysis. Physica A: Statistical Mechanics and its Applications, 460, 222–229. Cattani, C., Pierro, G. and Altieri, G. (2012) Entropy and multifractality for the myeloma multiple TET 2 gene. Mathematical Problems in Engineering. Cattani, C. (2010) Fractal patterns in prime numbers distribution. International Conference on Computational Science and Its Applications, 164–176.

page 206

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Bibliography

b4547-main

207

Cattani, C. (2010) Fractals and hidden symmetries in DNA. Mathematical Problems in Engineering 2010. Cattani, C. (2020) Cantor waves for Signorini hyperelastic materials with cylindrical symmetry. Axioms, 9(1). Cattani, C., Ehler, M., Li, M., Liao, Z. and Hooshmandasl, M. (2014) Scaling, selfsimilarity, and systems of fractional order. Abstract and Applied Analysis. Cattani, C. and Laserra, E. (2010) Self-similar hierarchical regular lattices. International Conference on Computational Science and Its Applications, 225–240. Cattani, C., Laserra, E. and Bochicchio, I. (2012) Simplicial approach to fractal structures. Mathematical Problems in Engineering, 1–21. Cawley, R. and Mauldin, R. D. (1992) Multifractal decomposition of Moran fractals. Adv. Math., 92, 196–236. Chen, Z. Y., Cattani, C. and Zhong, W. P. (2014) Signal processing for nondifferentiable data defined on Cantor sets: A local fractional Fourier series approach. Advances in Mathematical Physics. Chen, G. and Cheng, Q. (2017) Fractal density modeling of crustal heterogeneity from the KTB deep hole. J. Geophys. Res. Solid Earth, 122, 1919–1933. Doi:10.1002/2016JB013684. Choquet, G. (1969) Lectures on Analysis. Vol II. Benjamin. New York. Chu, P. C. (1999) Multifractal analysis of the southwestern iceland sea surface mixed layer thermal structure. Thirteenth Symposium on Boundary Layers and Turbulence, American Meteorological Society, 476–479. http://hdl.handle.net/10945/36212. Cole, J. (2000) Relative analysis. Choas Solitons Fractals, 11, 2233–2250. Cole, J. and Olsen, O. (2003) Multifractal variation measures and multifractal density theorems. Real. Anal. Exch., 28, 501–514. Colebrook, C. M. (1970) The Hausdorff dimension of certain sets of nonnormal numbers. Michigan Math. J., 17, 103–116. Collet, P., Lebowitz, J. and Porzio, A. (1987) The dimension spectrum of some dynamical systems. J. Stat. Phys, T., 47, 609–644. Dai, D. (1995) Frostman lemma for Hausdorff measure and packing measure in probability spaces. J Nanjing Univ Math Biquarterly, 12, 191–203. Dai, C. (2008) On the equivalence of the multifractal centered Hausdorff measure and the multifractal packing measure. Nonlinearity, 21, 1443–1453. Dai, M. and Jiang, Y. (2009) The equivalence of the multifractal on cookie-cutterlike sets. Chaos Solitons Fractals, 41, 1408–1415. Dai, C. and Li, Y. (2006) A multifractal formalism in a probability space. Chaos Solitons Fractals, 27, 57–73. Dai, M. and Liu, Z. (2008) The quantization dimension and other dimensions of probability measures. Int. J. Nonlinear Sci., 5, 267–274. Das, M. (1998) Local properties of self-similar measures. Illinois J. Math., 42, 313–332. Das, M. (2005) Hausdorff measures, dimensions and mutual singularity. Trans. Am. Math. Soc., 357, 4249–4268.

page 207

February 7, 2022 16:4

208

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Bibliography

David, G. and Semmes, S. (1997) Fractured Fractals and Broken Dreams. Clarendon Press, Oxford. Debussche, A. (1998) Hausdorff dimension of random invariant set. JPAM., 77, 967–988. Dolotin, V. and Morozov, A. (2006) Universal Mandelbrot Set: Beginning of the Story, World Scientific Publishing Company. Dryakhlov, A. V. and Tempelman, A. A. (2001) On Hausdorff dimension of random fractals. New York J. Math., 7, 99–115. Edgar, G. A. (1998) Integral, Probability and Fractal Measures. Springer-Verlag New York. Edgar, G. A. (2008) Measures, Topology and Fractal Geometry. Undergraduate texts in Maths. Undergraduate texts in mathematics, Springer-Verlag. Eggleston, H. G. (1949) The fractional dimension of a set defined by decimal properties. J. Math., 20, 31–36. Eggleston, H. G. (1951) The fractional dimension of a set defined by decimal properties. J. Math., 20, 31–36. Falconer, K. J. (1994) The multifractal spectrum of statistically self-similar measures. J. Theor. Probab., 7(3), 681–702. Falconer, K. J. (1985) The geometry of fractal sets. Lecturer in Mathematics, University of Bristol. Falconer, K. J. (1990) Fractal Geometry. John Wiley and Sons. Kenneth, J. Falconer (auth.), Christoph Bandt, Siegfried Graf, Martina Z¨ ahle (1995) Fractal Geometry and Stochastics, Series: Progress in Probability, Volume 37, Birkh¨ auser Basel, Boston. Fan, A. H. and Feng, D. J. (1998) Analyse multifractale de la r´ecurence sur l’espace symbolique. CRAS. Paris, t. 327, S´erie, 1, 629–632. Fan, Q., Liu, S. and Wang, K. (2019) Multiscale multifractal detrended fluctuation analysis of multivariate time series. Physica A, 532, 121864. DOI: 10.1016/j.physa.2019.121864. Fernandez-Martinez, M., Guirao, J. L. G., Sanchez-Granero, M. A. and Segovia, J. E. T. (2019) Fractal Dimension for Fractal Structures: With Applications to Finance. SEMA SIMAI Springer Series 19, Springer International Publishing. ISBN: 978-3-030-16644-1;978-3-030-16645-8. de Figueiredo, B. C. L., Moreira, G. R., Stosic, B. and Stosic, T. (2014) Multifractal analysis of hourly wind speed records in petrolina, northeast brazil. Rev. Bras. Biom., Sao Paulo, 32(4), 599–608. Fillol, J. (2005) Mod´elisation multifractale du taux de change dollar/euro. Economie Internationale, (104), 135–150. Flake, G. W. (1998) The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation. MIT Press (MA). Frostman, O. (1935) Potentiel d’´equilibre et capacit´e des ensembles avec quelques applications a ` la th´eorie des fonctions. Th`ese de Doctorat de la Facult´e des Sciences de Lund. Frame, M. and Cohen, N. (2015) Fractals and Dynamics in Mathematics, Science, and the Arts: Theory and Applications Benoit Mandelbrot: A Life in Many Dimensions, World Scientific.

page 208

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Bibliography

b4547-main

209

Frisch, U. and Parisi, G. (1985) Fully developed turbulence and intermittency, Proc. Int. Summer school Phys., Enrico Fermi, North Holland, 84–88. Garcia, I. and Zuberman, L. (2012) Exact packing measure of central Cantor sets in the line. Journal of Mathematical Analysis and Applications, 386(2), 801–812. Garcia, I., Molter, U. M. and Scotto, R. (2007) Dimension functions of Cantor sets. Proc. Am. Math. Soc., 135(10), 3151–3161. Gatzouras, D. (1999) Lacunarity of self-similar and stochastically self-similar sets. Trans of AMS, 352(5), 1953–1983. Gervais, J.-J. (2009) Introduction aux fractals et aux syst`emes dynamiques. 2e version, Aoˆ ut. Guo, Q. Jiang, H. and Xi, L. (2006) Hausdorff Dimension of Generalized Sierpinski Carpet. Interna. J. Nonlinear Sci, 6, 153–158. Gurevich, B. M. and Tempelman, A. A. (1999) Hausdorff dimension and thermodynamic formalism. Communication of the Moscow Mathematical Society. Guzman, M. (1975) Differentiation of Integrals on Rd . Lecture Notes in Mathematics. N 481. Springer Verlag. Halsey, T. C., Jensen, M. H., Kadano, L. P., Procaccia, I. and Shraiman, B. I. (1986) Fractal measures and their singularities: The characterization of strange sets. Phys. Rev. A, 33, 1141–1151. Hare, K., Mendivil, F. and Zuberman, L. (2013) The sizes of rearrangements of Cantor sets. Cana. Math. Bull., 56(2), 354–365. Hare, K. and Zuberman, L. (2010) Classifying Cantor sets by their multifractal spectrum. Nonlinearity, 23(11), 2919–2933. Heurteaux, Y. (1998) Estimations de la dimension inf´erieure et de la dimension sup´erieure des mesures. Ann. Ins. H. Poicar´e. Probab. Statist., 34, 309–335. Huang, X. and El-Sayed, M. A. (2010) Gold nanoparticles: Optical properties and implementations in cancer diagnosis and photothermal therapy. Journal of Advanced Research, 1(1), 13–28. Hudson, R. L. and Mandelbrot, B. B. (2005) The Misbehavior of Markets: A Fractal View of Financial Turbulence: A Fractal View of Risk, Ruin, and Reward, Basic Books. Hutchinson, J. (1981) Fractals and self-similarity. Indiana Univ. Math. J., 30, 713–747. Jarvenpaa, E., Jarvenpaa, M., Kaenmaki, A., Rajala, T., Rogovin, S. and Suomala, V. (2017) Packing Dimension and Ahlfors Regularity of Porus Sets in Metric Spaces. arXiv:1701.08593v1 [math.CA] 30 Jan 2017. Karaca, Y. and Cattani, C. (2017) Clustering multiple sclerosis subgroups with multifractal methods and self-organizing map algorithm. Fractals, 25(04), 1740001. Karaca, Y., Cattani, C., Moonis, M. and Bayrak, S. (2018) Stroke subtype clustering by multifractal bayesian denoising with fuzzy means and-means algorithms. Complexity. Khalili, G. A. and Cattani, C. (2019) Fractal logistic equation. Fractal and Fractional, 3(3), 41.

page 209

February 7, 2022 16:4

210

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Bibliography

Kumar, S., Nisar, K. S., Kumar, R. Cattani, C. and Samet, B. (2020) A new Rabotnov fractional-exponential function-based fractional derivative for diffusion equation under external force. Math. Meth. Appl. Sci., 43(7), 1–12. King, J. (1995). The singularity spectrum for general Sierpinski carpets. Adv. Math., 116, 1–11. Kigami, J. (2001). Analysis on Fractals. Cambridge Tracks in Mathematics. Cambridge University Press. Lambert, A. (2012) Th´eorie de la mesure et int´egration, Exercices et Corrig´es, Licence de Math´ematiques L3, UE LM364 & LM365, Universit´e Pierre et Marie Curie (Paris 6), Ann´ee 2012–2013. Li, M., Zhao, W. and Cattani, C. (2013) Delay bound: fractal traffic passes through network servers. Mathematical Problems in Engineering. Liu, K., Hu, R. J., Cattani, C., Xie, G. N., Yang, X. J. and Zhao, Y. (2014) Local Fractional-Transforms with Applications to Signals on Cantor Sets. Abstract and Applied Analysis. Li, M., Hui, X. F., Cattani, C., Yang, X. J. and Zhao, Y. (2014) Approximate solutions for local fractional linear transport equations arising in fractal porous media. Advances in Mathematical Physics. Liu, S., Cattani, C. and Zhang, Y. (2019) Introduction of fractal based information processing and recognition. Appl. Sci., 9(7), 1297. Lucas, A. (2000) Mesure de Hausdorff d’ensembles li´es aux oscillations de Wienner. CRAS. Paris, t. 330, S´erie 1, 509–512. Makarov, N. G. (1985). On the distortion of boundary sets under conformal mappings. Proc. Lond. Math. Soc., 51, 369–384. Mandelbrot, B. B. (1974) Intermittent turbulence in self-similar cascades divergence of high moments and dimension of the carrier. J. Fluid Mech., 62, 331–358. Mandelbrot, B. B. (1982) The Fractal Geometry of Nature, 1st edn., W. H. Freeman and Company. Mandelbrot, B. B. (1993). Negative dimensions and H¨ olders, multifractals, and the role of lateral preasymptotics in science. Fourier Analysis and Applications, Paris. Edited by J. Peyri`ere et J. P. Kahane meeting. Mandelbrot, B. and Ried, R. (1995). Multifractal formalism for infinite multinomial measures. Adv. Appl. Math., 16, 132–150. Mandelbrot, B. B. (1997) Fractals and Scaling in Finance. Springer. Mandelbrot, B. B. (1999) Les objets fractals: forme, hasard et dimension, survol du langage fractal, 4th edn., Flammarion. Mandelbrot, B. B. (2004) Fractals and Chaos: The Mandelbrot Set and Beyond, 1st ed., Springer-Verlag New York. Mandelbrot, B. B. and Hudson, R. L. (2006) The Misbehavior of Markets: A Fractal View of Financial Turbulence, Basic Books. Mandelbrot, B. B. (2012) The Fractalist, Memoir of a Scientific Maverick Pantheon. Meneveau, C., Sreenivasan, K. R., Kailasnath, P. and Fan, M. S. (1990) Joint multifractal measures: Theory and applications to turbulence. Physical Review A, 41(2), 894–913.

page 210

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Bibliography

b4547-main

211

Mattila, P. (1995) Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability. Cambridge University Press. Mauroy, B., Filoche, M., Weibel, E. R. and Sapoval, B. (2004) An optimal bronchial tree may be dangerous. Nature, 427, 633–636. Mearns, B. (2021) Fractal Fern (https://www.mathworks.com/matlabcentral/ fileexchange/4372-fractal-fern), MATLAB Central File Exchange. Retrieved April 12, 2021. Menceur, M., Ben Mabrouk, A. and Betina, K. (2016) The multifractal formalism for measures, review and extension to mixed cases. Anal. Theory Appl., 32(1), 77–106. Menceur, M. and Ben Mabrouk, A. (2019) A joint multifractal analysis of vector valued non Gibbs measures. Chaos, Solitons Fractals, 126, 203–217. Mignot, P. (1998) Une m´ethode d’estimation du spectre multifractal, CRAS. Paris, 327(I), 689–692. Ngai, S. M. (1997) A dimension result arising from the Lq -spectrum of a measure. Proc. AMS, 125(10), 2943–2951. Nguyen, N. (2001) Iterated function systems of finite type and week separation property. Proc. AMS., 130(2), 483–487. Novak, M. M. (2004) Thinking in Patterns: Fractals and Related Phenomena in Nature. World Scientific. Olivier, E. (1998) Analyse multifractale de fonctions continues, CRAS. Paris, 326(I), 1171–1174. Olsen, L. (1995) A multifractal formalism. Adv. Math., 116, 82–196. Olsen, L. (1996) Multifractal dimensions of product measures. Math. Proc. Camb. Phil. Soc., 120, 709–34. Olsen, L. (2000) Dimension inequalities of multifractal Hausdorff measures and multifractal packing measures. Math Scand, 86, 109–29. Olsen, L. (2003) Mixed divergence points of self-similar measures. Indiana Univ. Math. J., 52, 1343–1372. Olsen, L. (2004) Mixed generalized dimensions of self-similar measures. J. Math. Anal. Appl., 306, 516–539. Olsen, L. (auth.), Christoph Bandt, Siegfried Graf, Martina Z¨ ahle (2000) Fractal Geometry and Stochastics II, Series: Progress in Probability (ed.), Vol. 46, Birkh¨ auser Basel, Boston. O’Neil, T. C. (1997) The multifractal spectrum of quasi self-similar measures. J. Math. Anal. Appl., 211, 233–257. Patzschke, N. (1997) The strong open set condition in the random case. Proc. AMS., 125(7), 2119–2125. Pesin, Y. B. (1997) Dimension Theory in Dynamical Systems. Contemporary Views and Applications. Chicago Lectures in Mathematics. Pesin, Y. B. and Climenhaga, V. (2009) Lectures on Fractal Geometry and Dynamical Systems. Students Mathematical Library, Vol. 52, Mathematics Advanced Study, American Mathematical Society. Peitgen, H.-O., Saupe, D., Fisher, Y., McGuire, M., Voss, R. F., Barnsley, M. F., Devaney, R. L. and Mandelbrot, B. B. (1988) The science of fractal images, 1st edn., Springer-Verlag.

page 211

February 7, 2022 16:4

212

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Bibliography

Peitgen, H.-O., Jurgens, H. and Saupe, D. (1992), Chaos and Fractals. New Frontiers of Science. Springer Verlag. Peitgen, H.-O., Jurgens, H. and Saupe, D. (1992) Fractals for the Classroom: Part Two: Complex Systems and Mandelbrot Set, 1st edn., Springer-Verlag, New York. Peyri`ere, J. (1992) Multifractal measures. Dordrecht Kluwer Academic Press, ed Proceeding of the NATO advanced study institute on probabilistic and stochastic methods in analysis with applications, Vol. 372. Pickover, C. A. (1995) The Pattern Book: Fractals, Art, and Nature. World Scientific. Rashid, T. (2014) Make Your Own Mandelbrot, CreateSpace Independent Publishing Platform. Rand, D. A. (1989) The singularity spectrum f (α) for Cookie-Cutters, Ergod. Th. and Dynam. Sys., 9, 527–541. Raymond, X. S. and Tricot, C. (1988) Packing regularity of sets in n-space. Math. Proc. Camb. Phil. Soc., 103, 133–145. Redondo, J. M. (1993) Fractal models of density interfaces. IMA. Conf. Ser. 13, (ed. M. Frage, JCR, Hunt, JC Vassilicos), 353–370, Elsevier. Riedi, R. (1995) An improved multifractal formalism and self-similar measures. J. Math. Anal. Appl., 189, 527–541. Riedi, R. H. and Scheuring, I. (1997) Conditional and relative multifractal spectra. Fractals: An Interdisciplinary J., 5, 153–68. Robert, L.-H. (2020) Th´eorie de la mesure et int´egration. Universit´e de Gen`eve, Printemps, Section de Math´ematiques. Rogers, C. A. (1970) Hausdorff Measures, Cambridge University Press. Sapoval, B. and Filoche, M. (2010) Pourquoi le poumon est-il aussi robuste? La Recherche, F´evrier. Scheuring, I. and Riedi, R. H. (1994) Application of multifractals to the analysis of vegetation pattern. Journal of Vegetation Science, 5, 489–496. Scholz, C. H. and Mandelbrot, B. B. (1989) Fractals in Geophysics. In Pure and Applied Geophysics, 1 ed., Birkh¨ auser Basel. Selezneff, A. (2011) Crit`eres de capacit´e nulle. M´emoire de Maˆıtrise en Math´ematiques, D´epartement de Math´ematiques et de Statistique, Facult´e des Sciences et de G´enie, Universit´e Laval, Qu´ebec. Sierpinski, W. (1916) Sur une courbe cantorienne qui contient une image biunivoque et continue de toute courbe donn´ee. C. R. Acad. Sci. Paris (in French). 162, 629–632. Spear, D. W. (1992) Measures and self-similarity. Adv. Math., 91, 143–157. Stauffer, D. and Stanley, E. H. (1991) From Newton to Mandelbrot: A Primer in Modern Theoretical Physics, 2nd edn., Springer-Verlag. Taylor, S. J. (1995) The fractal analysis of Borel measures in Rd . J. Fourier. Anal. Appl., Kahane Special Issue, 553–568. Triebel, H. (1992) Theory of function spaces II. Birkh¨ auser Verlag. Basel. Boston, Berlin.

page 212

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Bibliography

b4547-main

213

Veerman, P. (1998) Hausdorff Dimension of Boundaries of Self-Affine Tiles in Rn . Bol. Mex. Mat, 34, 1–24. Vehel, J. L. and Legrand, P. (2004) Signal and image processing with FRACLAB, DOI: 10.1142/9789812702746− 0032, FracLab, Available http://www-rocq. inria.fr/fractales. Vehel, J. L. and Vojak, R. (1998) Multifractal analysis of Choquet capacities. Adv. Appl. Math., 20, 1–43. Walter, C. (2001), Les Echelles de Temps sur les March´es Financiers. Revue de Synth`ese, 4(1), 55–69. Wang, L. F., Yang, X. J., Baleanu, D., Cattani, C. and Zhao, Y. (2014) Fractal dynamical model of vehicular traffic flow within the local fractional conservation laws. Abstract and Applied Analysis. Wanqing, S., Chen, X., Cattani, C. and Zio, E. (2020) Multifractional Brownian Motion and Quantum-Behaved Partial Swarm Optimization for Bearing Degradation Forecasting. Complexity. Weibel, E. R. (1963) Morphology of the Human Lung, Springer Verlag and Academic Press, Heidelberg-New York. Wu, J. M. (1998) Hausdorff dimension on metric spaces. Proc. AMS, 126(5), 1453–1459. Wu, M. (2005) The multifractal spectrum of some Moran measures. Science in China Series A: Mathematics Volume, 48, 1097–1112. Doi: 10.1360/ 022004-10. Wu, M. (2005) The singularity spectrum f (α) of some Moran fractals. Monatsh Math, 144, 141–155. Wu, M. and Xiao, J. (2011) The singularity spectrum of some non-regularity Moran fractals. Chaos Solitons Fractals, 44, 548–557. Xiao, J. and Wu, M. (2008) The multifractal dimension functions of homogeneous Moran measure. Fractals, 16, 175–185. Xiaojun, Y., Zhao, Y., Baleanu, D., Cattani, C. and Cheng, D. F. (2013) Maxwell’s Equations on Cantor Sets: A Local Fractional Approach. Xu, S., Ling, X., Cattani, C., Xie, G. N., Yang, X. J. and Zhao, Y. (2014) Local fractional Laplace variational iteration method for nonhomogeneous heat equations arising in fractal heat flow. Mathematical Problems in Engineering. Xu, M. and Wang, S. (2011) The boundedness of bilinear singular integral operators on Sierpinski Gaskets. Anal. Theory Appl., 27, 92–100. Xu, S. and Xu, W. (2012) Note on the Paper, An Negative Answer to a Conjecture on the Self-similar Sets Satisfying the Open Set Condition. Anal. Theory Appl., 28, 49–57. Xu, S., Xu, W. and Zhong, D. (2012) Some new iterated function systems consisting of generalized contractive mappings. Anal. Theory Appl., 28, 269–277. Yang, A. M., Cattani, C., Zhang, C., Xie, G. N. and Yang, X. J. (2014) Local fractional Fourier series solutions for nonhomogeneous heat equations

page 213

February 7, 2022 16:4

214

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Bibliography

arising in fractal heat flow with local fractional derivative. Advances in Mechanical Engineering, 6, 514639. Yang, X. J., Cattani, C. and Xie, G. (2015) Local fractional calculus application to differential equations arising in fractal heat transfer. Fract. Dyn., 272–285. Yang, X. J., Machado, J. A. T., Cattani, C. and Gao, F. (2017) On a fractal LCelectric circuit modeled by local fractional calculus. Commun. Nonlinear Sci. Numer. Simul., 47, 200–206. Yang, X. J., Srivastava, H. M. and Cattani, C. (2015) Local fractional homotopy perturbation method for solving fractal partial differential equations arising in mathematical physics. Rom. Rep. Phys., 67(3), 752–761. Yang, A. M., Zhang, Y. Z., Cattani, C., Xie, G. N., Rashidi, M. M., Zhou, Y. J. and Yang, X. J. (2014) Application of local fractional series expansion method to solve Klein-Gordon equations on Cantor sets. Abstr. Appl. Anal., 55. Yang, A. M., Zhang, C., Jafari, H., Cattani, C. and Jiao, J. (2014) Picard successive approximation method for solving differential equations arising in fractal heat transfer with local fractional derivative. Abstr. Appl. Anal., 2014. Yan, S. H., Chen, X. H., Xie, G. N., Cattani, C. and Yang, X. J. (2014) Solving Fokker-Planck equations on Cantor sets using local fractional decomposition method. Abstr. Appl. Anal. Ye, Y.-L. (2007) Self-similar vector-valued measures. Adv. Appl. Math., 38, 71–96. Yeh, J. (2014) Real Analysis: Theory of Measure and Integration. World Scientific. Yu, Q., Dave, R. N., Zhu, C., Quevedo, J. A. and Pfeffer, R. (2005) Enhanced fluidization of nanoparticles in an oscillating magnetic field. AIChE J., 51(7), 1971–1979. Yuan, Y. (2015) Spectral self-affine measures on the generalized three sierpinski gasket. Anal. Theory Appl., 31, 394–406. Zhang (2018) Physical fundamentals of nanomaterials. Micro & Nano Technologies Series, Chemical Industry Press, William Andrew Applied Sciences Publisher. Elsevier. Zhang, Y., Cattani, C. and Yang, X. J. (2015) Local fractional homotopy perturbation method for solving non-homogeneous heat conduction equations in fractal domains. Entropy, 17(10), 6753–6764. Zeng, C., Yuan, D. and Xui, S. (2012) The hausdorff measure of sierpinski carpets basing on regular pentagon. Anal. Theory Appl., 28, 27–37. Zhou, Z. and Feng, L. (2012) A theoretical framework for the calculation of hausdorff measure self-similar set satisfying OSC. Anal. Theory Appl., 27, 387–398. Zhu, Z. and Zhou, Z. (2014) A local property of hausdorff centered measure of self-similar sets. Anal. Theory Appl., 30, 164–172.

page 214

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Index

c-adic interval, 74, 91, 93, 95 s-covering, 129

S-invariant set, 80 α-H¨ older, 67 α-H¨ olderian, 67 α-singularity set, 159 δ-covering, 135, 137, 138, 140, 141, 145, 148, 151, 157, 158 δ-packing, 136, 140–142, 146–148, 151–153, 155–157, 167 -covering, 50–54, 56, 57, 61, 63, 65, 67, 68, 168, 176 -packing, 101, 104, 107, 120, 176 η-covering, 160 η-packing, 161 A-measurable, 27, 32 A-measurable function, 27 B-measurable, 27, 32, 38 B-measurable function, 27 B∞ -measurable, 40, 46, 49 Bν -measurable, 40 Bn -measurable, 38, 40 Hα -measurable, 53 Hα -measurable set, 53, 54 µ-dimension, 91 µ-regular, 14 σ-additive, 24 σ-algebra, 6, 8, 9, 11, 13, 15, 16, 22, 23, 38, 39, 54, 89 σ-field, 27–32, 35, 47, 48 σ-stability, 143, 144 σ-stable, 143

Additive, 97, 136 Additivity, 136, 138, 144 Adherance value, 130 Algorithm, 179, 182 Alphabet, 123 Analysis, 1 Annulus, 12 Application, viii, 4, 5, 179, 180, 191 Applied, 3, 4 Area, 183 Asymptotic behaviour, 132 Attractor, 114, 182 Ball, 24, 56, 66–68, 71, 74, 75, 86, 87, 101, 102, 116, 136, 151, 191 Banach space, 36 Barnsley, 181 Barnsley fern, 181, 182 Barnsley numerical fern, 182 Basic, 4, 6 Basis, 20, 184 Behaviour, 174, 189, 190 Besicovitch covering theorem, 140, 150, 157, 158, 170 Billingsley theorem, 122, 169 Borel, 134 Borel σ-algebra, 9, 14, 17, 18, 23 Borel finite measure, 133 215

page 215

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

216

Borel measure, 67, 71, 75, 89 Borel probability measure, 68, 79, 85–87, 118, 121, 124, 125, 127, 135, 149, 159, 166, 175, 177, 178 Borel set, 12, 13, 17, 53, 67, 85, 87, 92, 125, 133, 176, 177 Borel-Cantelli lemma, 163, 164, 166, 174 Bouligand–Minkowski dimension, 100, 103, 134, 191 Boundary, 25 Bounded, 16, 17, 60, 67, 81, 83, 105, 106, 108, 110–113, 121 Bounded function, 16, 18 bounded measure, 9 Bounded sequence, 96 Bounded set, 25, 66, 84, 90, 101, 112 Box, 129, 134, 191, 193 Box counting dimension, 200, 201 Box dimension, 100, 103, 114, 116, 191, 196 Box-counting, 193 Cantor, 3, 57, 196 Cantor set, 57, 76, 98, 128–130, 194, 197, 200 Capacity, 77, 97, 98, 185 Capacity dimension, 4, 77, 78, 84 Caratheodory, 3 Cauchy sequence, 36, 89 Cauchy-Schwartz inequality, 165 Circle, 191 Closed, 8, 13, 17, 28, 30, 60, 63, 73 Closed ball, 101, 102 Closed interval, 60 Closed set, 14–16, 18, 23, 98 Coarse, 6 Coarse topology, 8 Coefficient, 193 Collection, 6, 7, 9, 15, 18, 80, 118, 137 Compact, 18–21, 84, 97 Compact set, 21, 51, 68, 72, 75, 76, 78, 80, 81, 83, 98, 129, 182, 196 Complement, 6, 7, 15 Complementary, 8, 17 Complete metric space, 51, 80

b4547-main

Index

Complete separable metric space, 18 Computation, 3, 197 Concave, 118, 131, 132 Concavity, 119 Conditional expectation, 31, 32, 34, 41 Constant function, 30, 32 Continuity set, 16 Continuous, 25, 71, 73, 129, 130 Continuous function, 16, 18, 132 Contraction, 80, 82–84, 97, 197 Contractive, 48, 80, 83 Contractive linear operator, 29, 48 Contractive self-similarity, 85 Contractive similarity, 182 Contractive similitude, 197 Contractive transformation, 184 Convergence, 33, 35, 96 Convergent, 12, 18, 43, 44, 46, 82, 83, 96, 166 Convergent weakly, 19 Convex, 118, 132, 145, 164, 165 Convex function, 133 Convex set, 132 Convexity, 119, 146, 147 Countable, 20, 50–52, 60, 101, 104, 110 Countable collection, 6–8 Countable partition, 117 Countable set, 20, 21, 55, 138 Countable union, 7, 15 Covering, 13, 68, 69, 99, 108, 118, 137, 139, 145, 146, 150, 151, 155, 162, 175 Cube, 13, 54, 69, 70, 98, 99, 102, 133, 191 Decreasing, 18, 41, 118, 131, 132, 135, 145, 151, 154, 163, 176, 177 Decreasing function, 63, 136 Decreasing sequence, 24, 31, 64, 83 Dense, 19 Density, 86, 90 Density measure, 96 Development, vii, viii, 4, 5

page 216

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

Index

Diameter, 50, 66, 77, 98, 101, 116, 132, 136, 193 Difference, 7, 27 Differentiable, 121, 122, 131, 164, 166, 169, 174, 197 Differentiable function, 199 Dimension, 3–5, 13, 54, 57, 63, 74, 76, 77, 100, 110, 114, 118, 129, 142, 194, 196 Dimension theory, 2 Discrete, 4 Discrete Hausdorff dimension, 99 Discrete topology, 8 Disjoint, 64, 65, 70, 86, 101 Disjoint balls, 115 Disjoint sets, 16, 17 Distance, 31, 51, 72, 80–82, 97, 98 Distribution, 183 Dominated convergence, 36 Dyadic, 70 Dyadic cube, 69–71, 101, 193 Dyadic interval, 74 Dynamical system, 180 Dynamical system, 2 Element, 6–8, 10, 14, 20, 21, 23, 24, 63, 86, 118, 123, 130, 151 Equality, 152, 159, 168 Equi-integrability, 37 Equi-integrable, 35 Equi-integrable sequence, 36 Equivalent, 16, 25 Error measure, 187 Euclidean distance, 101, 132 Euclidean geometry, 2 Euclidean norm, 50, 77 Euclidean space, 2, 3, 6, 28, 50, 118, 184 Exercise, viii, 23–25, 47–49, 74–76, 97, 98, 113–115, 129–133, 174–177, 199–201 Expectation, 38 Experimental, 179, 190, 191 Extension, 2, 5

b4547-main

217

Fatou’s Lemma, 11 Field, vii, 3, 179, 186, 191 Financial index, 186 Finite, 9, 14, 15, 20, 42, 43, 45, 55, 63, 70, 95, 98, 118, 122, 123, 127, 132, 148, 151, 152, 164, 166 Finite collection, 7 Finite intersection, 7, 8 Finite measure, 14–18, 25, 70, 133 Finite partition, 130 Finite set, 8, 80 Finite union, 7, 10, 20 Formalism, 3, 116 Fractal, vii, viii, 1–4, 100, 116, 179–181, 186, 189–191, 194, 196, 200, 201 Fractal analysis, vii, viii, 3–6, 27, 50, 100, 116, 179, 181, 186, 190 Fractal behavior, 186 Fractal dimension, 2–4, 77, 134, 189, 191, 194, 196 fractal dimension, 193 Fractal geometry, vii, 4, 59, 185, 186, 190 Fractal law, 186 Fractal measure, 4, 134 Fractal model, 3, 184, 189 Fractal modeling, 186 Fractal set, 3, 194 Fractal structure, 181, 183, 186, 188, 191 Fractality, 188 Fractional calculus, 190 Frostman function, 130 Frostman, 68 Function, 3, 16, 25, 27–29, 31, 36, 51, 67, 73, 75, 80, 116, 118, 124, 129–132, 134, 135, 159, 164–166, 175, 176, 187, 189, 197 Functional, 186 Functional space, 28 Functional theory, 3 General, 118 Generalization, 5, 134 Generalized, 129, 159, 163

page 217

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

218

Geometric, 183 Geometric structure, 184 Geometrical, 181 Geometry, 1–4, 6, 50, 100, 116, 179, 181 Gibbs, 5, 68 Gibbs measure, 5, 116, 117, 121, 128, 134, 159, 163 H¨ older regularity, 189 H¨ olderian, 67, 87 H¨ olderian measure, 67 H¨ older inequality, 48, 147 Hausdorff, 3, 5, 77, 84 Hausdorff dimension, 2, 4, 50, 55, 57, 63, 64, 66, 67, 74, 77, 85, 97, 116, 134, 142, 159, 189, 196, 199 Hausdorff dimensions, 100 Hausdorff distance, 51, 52, 76, 81 Hausdorff measure, 4, 50, 54, 56, 57, 63, 68, 91, 118, 134, 135, 175 Hausdorff metric, 76 Hilbert space, 31 Idempotent, 48 Image, 189–193 Image processing, 189 Inclusion, 135 Increasing, 10, 11, 19, 28, 33, 34, 45, 51, 89, 118 Increasing function, 91 Increasing sequence, 10, 11, 13, 24, 29, 33, 38, 97 Independent, 95, 129 Index, 43, 67 Inequality, 120, 138, 140, 151, 152 Infinite, 8, 151, 152 Integer, 6, 25, 130 Integrable function, 40, 86 Integrable martingale, 33–35, 37, 38, 48 Integrable real random variable, 37 Integrable upper martingale, 35 Intersection, 98, 159

b4547-main

Index

Interval, 2, 17, 43, 57–63, 65, 72, 74, 75, 98, 114, 122, 124, 127, 128, 188, 197, 198, 200 Invariance set, 182 Invariant, 7, 8, 15, 20, 28, 32, 80, 84, 97, 182 Invariant set, 194, 197, 199 Investigation, 190 Irregular, 3, 116 Irregularity, 190 Iterated function system, 114, 130, 194, 200, 201 Iteration, 130 Kernel, 98, 99 Large, 37, 89, 134 Large deviation formalism, 134, 163, 164 Large deviation theorem, 167, 169, 174 Law, 95 Lebesgue, 96 Lebesgue measurable, 13 Lebesgue measure, 2, 3, 13, 25, 28, 55, 60, 61, 96, 101, 110, 196, 200, 201 Lebesgue outer measure, 13 Legendre transform, 118, 120, 133, 189 Length, 2, 98, 123 Level set, 134, 189, 197 Limit, 16, 86, 129, 135, 136, 145, 146, 176 Linear, 196 Linear form, 85, 86, 89 Linear positive form, 89 Linear space, 28 Local dimension, 159 Local lower dimension, 159 Local scaling invariance, 116 Locally finite, 24 Logarithmic, 188 Logarithmic index, 142 Low, 117 Lower α-singularity set, 159

page 218

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Index

Lower bound, 13, 51, 55, 56, 63, 67, 135, 169 Lung, 183–185 Lung model, 184 Mandelbrot, 2, 4, 184 Mandelbrot lung model, 185 Mandelbrot model, 184 Market indices, 186 Martingale, vii, 4, 27, 33, 39, 48 martingale, 46 Materials, 189–191 Mathematical, 3 Mathematical analysis, 2 Measurable, 9, 10, 12, 24, 38, 40 Measurable function, 26, 38 Measurable set, 9–11, 13 Measurable space, 9, 14, 23, 26, 49 Measure, 3, 5, 9, 14, 16, 19, 20, 24, 50, 68, 69, 96, 100, 114, 116–118, 121, 123, 129, 131, 134, 159, 163, 167, 169, 175, 186, 196–198 Measure theory, vii, 2, 4, 6, 9, 68 Measurable, 176, 177 Metric, 8, 9, 11, 52, 136 Metric measure, 138, 139 Metric outer measure, 24, 136 Metric space, 8, 9, 11, 14, 176 Middle-third Cantor set, 114, 115 Minimization problem, 130 Minkowski dimension, 98 Model, viii, 181, 184, 188 Modeling, 4, 180, 186 Monotone, 51, 135, 136, 142 Monotone convergence, 33 Monotony, 135, 143, 145, 146 Multi-index, 128, 130 Multifractal, 5, 116 Multifractal analysis, 68, 116, 117, 134, 135, 159, 163, 176, 177, 179, 186, 196, 197 Multifractal formalism, 5, 116, 117, 128, 134, 135, 163, 189 Multifractal generalization, 142 Multifractal geometry, 186 Multifractal measure, 5, 134

219

Multifractal set, 68 Multifractal spectrum, 159, 189, 190 Multifractal structure, 186, 188 Multifractality, 188 Multiplicative, 128 Nanoimage, 191 Nanomaterials, 189–192 Nanoparticle, 193 Nanoparticles, 191–193 nanoparticles, 191 Natural, 6, 8, 118 Necessary, 4, 124 Neighborhood, 20 Non-atomic probability measure, 91, 93 Nonlinear autoregressive model, 187 Normalized, 129 Number, 99, 116, 129, 130, 191, 193 Number theory, 180 Numerical Barnsley fern, 183 Open, 8, 17, 84, 90, 91, 119, 134 Open ball, 20, 101, 104 Open bounded interval, 174 Open interval, 61, 118, 127 Open set, 8, 14–16, 19–23, 53, 75, 86, 90, 91, 119 Open set condition, 76, 84, 85, 114, 197 Optimal, 116 Optimal objective function, 130 Order, 13, 116, 129, 135 Original, 4 Oscillating, 191 Oscillating behavior, 190 Oscillating behavior, 191 Outer, 63 Outer measure, 9–11, 22, 77, 108, 135, 136, 138, 139, 143, 147, 174 Outer metric measure, 12, 13, 51, 91, 108, 133, 176, 177 Outer regular measure, 10

page 219

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

220

Packing, 3, 114, 118, 142, 150, 175 Packing dimension, 4, 100, 110, 114, 134, 196 Packing measure, 4, 5, 100, 110, 114, 118, 134 Packing pre-measure, 118 Parameter, 174 Particles, 193 Partition, 118, 123 Partition function, 118, 188, 189 Phenomenon, 4 Porous structure, 190 Positive, 29, 32, 35, 40, 44, 47, 56, 99, 197 Positive capacity, 78 Positive continuous random variable, 44 Positive integrable martingale, 34, 35, 48 Positive integrable upper martingale, 48 Positive linear operator, 33, 48 Positive martingale, 41, 46 Positive measure, 9, 23 Positive random variable, 41, 43 Positive upper martingale, 41, 42, 45 Power moment, 133 Power set, 6 Pre-measure, 114 Probability, 129, 182 Probability measure, 9, 14, 18, 20, 23, 75, 77–79, 87, 124, 126, 129, 130, 198 Probability space, 27, 38, 95, 164 Probability theory, 38, 95, 134 Probability vector, 95, 196 Problem, viii, 4 Process, 58, 70, 161, 162, 194, 199 Projection, 31, 76 Prokorov theorem, 18 Property, 135, 136, 160, 161, 175, 191, 192 Pure, 2–4

b4547-main

Index

Quasi-Bernoulli, 128 Radius, 86 Random variable, 27, 38, 40, 42, 44, 46, 95, 164 Ratio, 84 Rational number, 24 Real analytic, 132 Real number, 13, 119, 131, 135, 145–147, 166, 174 Real random variable, 33, 35, 36, 44, 49 Real-valued, 26 Real-valued function, 118 Rectangle, 2, 3, 25, 94 Refinement, 9, 63, 72, 118, 130 Regression, 193 Regular, 11, 14–16, 18, 38, 39, 51, 53 Regular martingale, 39 Regular measure, 14 Regular metric measure, 63 Regular outer measure, 13 Regular stopping time, 33 Regularity, 33, 189 Relatively compact, 18, 19 Relatively compact measure, 18 Restriction, 13 Right continuous, 18 Right-open interval, 130 Scale, 2, 3, 117 Scaling function, 188–190 Scaling law, 3, 117, 183, 186, 188 Segment, 2, 181 Segmentation, 189 Self-similar, 3, 80, 117, 134, 181, 183, 194, 197 Self-similar function, 80 Self-similar measure, 196 Self-similar model, 181 Self-similar set, 196, 197 Semi-open interval, 63 Separable, 19, 20 Sequence, 9–11, 15, 16, 18–20, 23–25, 29, 33–35, 38, 40, 41, 43–47, 49, 55, 59, 61, 68, 69, 71, 81, 83, 86, 95, 96,

page 220

February 7, 2022 16:4

ws-book9x6

Fractal Analysis: Basic Concepts and Applications

b4547-main

Index

110, 118, 130, 136, 137, 139, 141, 143–145, 151, 154, 163, 164 Series, 12, 166, 186, 188–190 Set, 3, 4, 6–10, 13–16, 18, 21, 24, 42, 50–52, 55–57, 59, 63, 64, 66, 67, 72–77, 80, 81, 84, 85, 93, 95, 97, 98, 100, 103, 108, 110, 114–116, 123, 129, 132, 135–145, 147, 149, 151, 159–161, 165, 167–169, 174, 176, 186, 194, 196, 198, 200 Set function, 174, 175 Set theory, 3 Sierpinski, 196 Sierpinski carpet, 194–196 Sierpinski triangle, 199 Signal, 191 Signal processing, 189 Similarity, 84, 114 Similarity set, 197 Similitude, 197 Singularity, 117 Singularity set, 116, 117 Size, 129, 191, 193 Slope, 193 Solution, 130 Space, 28, 35, 36, 86, 117 Spectrum, 116, 117, 159, 197 Spectrum of singularities, 159 Spectrum of singularity, 116, 118, 159, 197 Spherical, 98 Square, 191 Statistical, 116, 117 Statistical measure, 189 Step function, 26 Stochastic, 187 Stopping time, 38–42, 49 Structure, 116, 180, 185 Subdivision, 17 Subin inequality, 46 Support, 71, 90, 116, 117, 145, 148, 149, 156–159, 163, 166–170, 173, 174, 177, 189, 197, 198 Surface, 2 Symmetric model, 184 System, 182

221

Theory, viii, 4, 31, 57, 63, 77, 196 Thermodynamical, 117 Thermodynamical potential, 116 Tight, 18, 20, 71 Time interval, 188 Time series, 186, 187, 189 Topological, 60 Topological dimension, 2 Topological interior, 118 Topological space, 8, 9 Topology, 8, 9 Total mass, 9 Transformation, 114, 181, 182, 184, 201 Triadic, 193 Triadic Cantor set, 57–60, 84, 97, 113, 194 Triadic interval, 85 Trivial, 8 Turbulence, 180 Ultra-metric distance, 73 Uniform, 129 Uniformly continuous, 89 Union, 7, 8 Unique, 84, 97, 118, 130, 182, 197 Unique solution, 84 Unit interval, 118 Upper, 159, 200, 201 Upper bound, 66, 104, 136 Upper bounded, 47 Upper dimension, 159 Upper limit, 165 Upper martingale, 33, 35 Upper positive martingale, 46 Value, 130 Variable, 118, 135, 136 Vector, 50, 77, 197 Vector space, 27 Volume, 13, 183, 186, 199 Wavelet, 190 Weakly, 16, 18 Weakly convergent, 126 Word, 123

page 221

B1948

Governing Asia

This page intentionally left blank

B1948_1-Aoki.indd 6

9/22/2014 4:24:57 PM

Series on Advances in Mathematics for Applied Sciences Editorial Board N. Bellomo Editor-in-Charge Department of Mathematics Politecnico di Torino Corso Duca degli Abruzzi 24 10129 Torino Italy E-mail: [email protected] M. A. J. Chaplain Department of Mathematics University of Dundee Dundee DD1 4HN Scotland C. M. Dafermos Lefschetz Center for Dynamical Systems Brown University Providence, RI 02912 USA J. Felcman Department of Numerical Mathematics Faculty of Mathematics and Physics Charles University in Prague Sokolovska 83 18675 Praha 8 The Czech Republic M. A. Herrero Departamento de Matematica Aplicada Facultad de Matemáticas Universidad Complutense Ciudad Universitaria s/n 28040 Madrid Spain S. Kawashima Department of Applied Sciences Engineering Faculty Kyushu University 36 Fukuoka 812 Japan

Alex - 12345 - Fractal Analysis.indd 1

F. Brezzi Editor-in-Charge IMATI - CNR Via Ferrata 5 27100 Pavia Italy E-mail: [email protected]

M. Lachowicz Department of Mathematics University of Warsaw Ul. Banacha 2 PL-02097 Warsaw Poland S. Lenhart Mathematics Department University of Tennessee Knoxville, TN 37996–1300 USA P. L. Lions University Paris XI-Dauphine Place du Marechal de Lattre de Tassigny Paris Cedex 16 France B. Perthame Laboratoire J.-L. Lions Université P. et M. Curie (Paris 6) BC 187 4, Place Jussieu F-75252 Paris cedex 05, France K. R. Rajagopal Department of Mechanical Engrg. Texas A&M University College Station, TX 77843-3123 USA R. Russo Dipartimento di Matematica II University Napoli Via Vivaldi 43 81100 Caserta Italy

5/1/2022 1:48:16 pm

Series on Advances in Mathematics for Applied Sciences Aims and Scope This Series reports on new developments in mathematical research relating to methods, qualitative and numerical analysis, mathematical modeling in the applied and the technological sciences. Contributions rlated to constitutive theories, fluid dynamics, kinetic and transport theories, solid mechanics, system theory and mathematical methods for the applications are welcomed. This Series includes books, lecture notes, proceedings, collections of research papers. Monograph collections on specialized topics of current interest are particularly encouraged. Both the proceedings and monograph collections will generally be edited by a Guest editor. High quality, novelty of the content and potential for the applications to modern problems in applied science will be the guidelines for the selection of the content of this series.

Instructions for Authors Submission of proposals should be addressed to the editors-in-charge or to any member of the editorial board. In the latter, the authors should also notify the proposal to one of the editors-in-charge. Acceptance of books and lecture notes will generally be based on the description of the general content and scope of the book or lecture notes as well as on sample of the parts judged to be more significantly by the authors. Acceptance of proceedings will be based on relevance of the topics and of the lecturers contributing to the volume. Acceptance of monograph collections will be based on relevance of the subject and of the authors contributing to the volume. Authors are urged, in order to avoid re-typing, not to begin the final preparation of the text until they received the publisher’s guidelines. They will receive from World Scientific the instructions for preparing camera-ready manuscript.

Alex - 12345 - Fractal Analysis.indd 2

5/1/2022 1:48:16 pm

Series on Advances in Mathematics for Applied Sciences Published*: Vol. 75

Applied and Industrial Mathematics in Italy II eds. V. Cutello et al.

Vol. 76

Geometric Control and Nonsmooth Analysis eds. F. Ancona et al.

Vol. 77

Continuum Thermodynamics by K. Wilmanski

Vol. 78

Advanced Mathematical and Computational Tools in Metrology and Testing eds. F. Pavese et al.

Vol. 79

From Genetics to Mathematics eds. M. Lachowicz and J. Miękisz

Vol. 80

Inelasticity of Materials: An Engineering Approach and a Practical Guide by A. R. Srinivasa and S. M. Srinivasan

Vol. 81

Stability Criteria for Fluid Flows by A. Georgescu and L. Palese

Vol. 82

Applied and Industrial Mathematics in Italy III eds. E. De Bernardis, R. Spigler and V. Valente

Vol. 83

Linear Inverse Problems: The Maximum Entropy Connection by H. Gzyl and Y. Velásquez

Vol. 84

Advanced Mathematical and Computational Tools in Metrology and Texting IX eds. F. Pavese et al.

Vol. 85

Continuum Thermodynamics Part II: Applications and Examples by B. Albers and K. Wilmanski

Vol. 86

Advanced Mathematical and Computational Tools in Metrology and Testing X eds. F. Pavese et al.

Vol. 87

Mathematical Methods for the Natural and Engineering Sciences (2nd Edition) by R. E. Mickens

Vol. 88

Road Vehicle Dynamics: Fundamentals of Modeling and Simulation by G. Genta and A. Genta

Vol. 89

Advanced Mathematical and Computational Tools in Metrology and Testing XI eds. A. B. Forbes et al.

Vol. 90

Advanced Mathematical and Computational Tools in Metrology and Testing XII eds. F. Pavese et al.

Vol. 91

Fractal Analysis: Basic Concepts and Applications by C. Cattani, A. B. Mabrouk and S. Arfaoui *To view the complete list of the published volumes in the series, please visit: https://www.worldscibooks.com/series/samas_series.shtml

Alex - 12345 - Fractal Analysis.indd 3

5/1/2022 1:48:16 pm