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Dimension and Recurrence in Hyperbolic Dynamics

  • Book
  • © 2008

Overview

Authors:
  1. Luis Barreira

    1. Departmento de Matemática, Instituto Superior Técnico, Lisboa, Portugal

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  • Pragmatic introduction to the study of dimension and recurrence in hyperbolic dynamics, traveling firmly but also rigorously from the basics to the frontiers of research in the area
  • More than half of the material appears here for the first time in book form, with the description of many recent developments in the area, on topics such as irregular sets, variational principles, applications to number theory, measures of maximal dimension, multifractal nonrigidity, and quantitative recurrence
  • The author is the winner of the Ferran Sunyer i Balaguer Prize 2008
  • Includes supplementary material: sn.pub/extras

Part of the book series: Progress in Mathematics (PM, volume 272)

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Table of contents (15 chapters)

  1. Front Matter

    Pages i-xiv
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  2. Introduction

    1. Introduction

      Pages 1-5

  3. Basic Notions

    1. Basic Notions

      Pages 7-15

  4. Dimension Theory

    1. Front Matter

      Pages 17-17
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    2. Dimension Theory and Thermodynamic Formalism

      Pages 19-39

    3. Repellers and Hyperbolic Sets

      Pages 41-66

    4. Measures of Maximal Dimension

      Pages 67-78

  5. Multifractal Analysis: Core Theory

    1. Front Matter

      Pages 79-79
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    2. Multifractal Analysis of Equilibrium Measures

      Pages 81-100

    3. General Concept of Multifractal Analysis

      Pages 101-125

    4. Dimension of Irregular Sets

      Pages 127-146

    5. Variational Principles in Multifractal Analysis

      Pages 147-164

  6. Multifractal Analysis: Further Developments

    1. Front Matter

      Pages 165-165
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    2. Multidimensional Spectra and Number Theory

      Pages 167-190

    3. Multifractal Rigidity

      Pages 191-208

    4. Hyperbolic Sets: Past and Future

      Pages 209-220

  7. Hyperbolicity and Recurrence

    1. Front Matter

      Pages 221-221
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    2. Pointwise Dimension for Hyperbolic Dynamics

      Pages 223-235

    3. Product Structure of Hyperbolic Measures

      Pages 237-254

    4. Quantitative Recurrence and Dimension Theory

      Pages 255-283
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Keywords

About this book

The main objective of this book is to give a broad uni?ed introduction to the study of dimension and recurrence inhyperbolic dynamics. It includes a disc- sion of the foundations, main results, and main techniques in the rich interplay of fourmain areas of research: hyperbolic dynamics, dimension theory, multifractal analysis, and quantitative recurrence. It also gives a panorama of several selected topics of current research interest. This includes topics on irregular sets, var- tional principles, applications to number theory, measures of maximal dimension, multifractal rigidity, and quantitative recurrence. The book isdirected to researchersas well as graduate students whowish to have a global view of the theory together with a working knowledgeof its main techniques. It can also be used as a basis for graduatecourses in dimension theory of dynamical systems, multifractal analysis (together with a discussion of several special topics), and pointwise dimension and recurrence in hyperbolic dynamics. I hope that the book may serve as a fast entry point to this exciting and active ?eld of research, and also that it may lead to further developments.

Authors and Affiliations

  • Departmento de Matemática, Instituto Superior Técnico, Lisboa, Portugal

    Luis Barreira

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