Thermodynamic Formalism : CIRM Jean-Morlet Chair, Fall 2019 / edited by Mark Pollicott, Sandro Vaienti.

Format:

Book

Contributor:

Pollicott, Mark, editor.

Vaienti, Sandro, editor.

Series:

Lecture Notes in Mathematics, 1617-9692 ; 2290

Language:

English

Subjects (All):

Dynamics.

Geometry, Differential.

Dynamical Systems.

Differential Geometry.

Local Subjects:

Dynamical Systems.

Differential Geometry.

Physical Description:

1 online resource (534 pages)

Edition:

1st ed. 2021.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2021.

Summary:

This volume arose from a semester at CIRM-Luminy on “Thermodynamic Formalism: Applications to Probability, Geometry and Fractals” which brought together leading experts in the area to discuss topical problems and recent progress. It includes a number of surveys intended to make the field more accessible to younger mathematicians and scientists wishing to learn more about the area. Thermodynamic formalism has been a powerful tool in ergodic theory and dynamical system and its applications to other topics, particularly Riemannian geometry (especially in negative curvature), statistical properties of dynamical systems and fractal geometry. This work will be of value both to graduate students and more senior researchers interested in either learning about the main ideas and themes in thermodynamic formalism, and research themes which are at forefront of research in this area.

Contents:

Intro

Foreword

Preface

Contents

Contributors

Part I Specifications and Expansiveness

1 Beyond Bowen's Specification Property

1.1 Introduction

1.2 Main Ideas: Uniqueness of the Measure of Maximal Entropy

1.2.1 Entropy and Thermodynamic Formalism

1.2.2 Bowen's Original Argument: The Symbolic Case

1.2.2.1 The Specification Property in a Shift Space

1.2.2.2 The Lower Gibbs Bound as the Mechanism for Uniqueness

1.2.2.3 Building a Gibbs Measure

1.2.3 Relaxing Specification: Decompositions of the Language

1.2.3.1 Decompositions

1.2.3.2 An Example: Beta Shifts

1.2.3.3 Periodic Points

1.2.4 Beyond Shift Spaces: Expansivity in Bowen's Argument

1.2.4.1 Topological Entropy

1.2.4.2 Expansivity

1.2.4.3 Specification

1.2.4.4 Bowen's Proof Revisited

1.3 Non-uniform Bowen Hypotheses and Equilibrium States

1.3.1 Relaxing the Expansivity Hypothesis

1.3.2 Derived-from-Anosov Systems

1.3.2.1 Construction of the Mañé Example

1.3.2.2 Estimating the Entropy of Obstructions

1.3.2.3 Specification for Mañé Examples

1.3.3 The General Result for MMEs in Discrete-Time

1.3.4 Partially Hyperbolic Systems with One-Dimensional Center

1.3.4.1 A Small Collection of Obstructions

1.3.4.2 A Good Collection with Specification

1.3.5 Unique Equilibrium States

1.3.5.1 Topological Pressure

1.3.5.2 Regularity of the Potential Function: The Bowen Property

1.3.5.3 The Most General Discrete-Time Result

1.3.5.4 Partial Hyperbolicity

1.4 Geodesic Flows

1.4.1 Geometric Preliminaries

1.4.1.1 Overview

1.4.1.2 Surfaces

1.4.1.3 Invariant Foliations via Horospheres

1.4.1.4 Jacobi Fields and Local Construction of Stables/Unstables

1.4.2 Equilibrium States for Geodesic Flows

1.4.2.1 The General Uniqueness Result for Flows.

1.4.2.2 Geodesic Flows in Non-positive Curvature

1.4.2.3 Uniqueness Can Fail Without a Pressure Gap

1.4.2.4 Uniqueness Given a Pressure Gap

1.4.2.5 Pressure and Periodic Orbits

1.4.2.6 Main Ideas of the Proof of Uniqueness

1.4.2.7 Unique MMEs for Surfaces Without Conjugate Points

1.4.2.8 Geodesic Flows on Metric Spaces

1.4.3 Kolmogorov Property for Equilibrium States

1.4.3.1 Moving Up the Mixing Hierarchy

1.4.3.2 Ledrappier's Approach

1.4.3.3 Decompositions for Products

1.4.3.4 Expansivity Issues

1.4.4 Knieper's Entropy Gap

1.4.4.1 Entropy in the Singular Set

1.4.4.2 Warm-Up: Shifts with Specification

1.4.4.3 Entropy Gap for Geodesic Flow

1.4.4.4 Other Applications of Pressure Production

References

2 The Role of Continuity and Expansiveness on Leo and Periodic Specification Properties

2.1 Introduction

2.2 Definitions

2.3 Proofs

2.3.1 Proof of Theorem 1.2

2.3.2 Proof of Theorem 1.3

2.3.3 Proof of Theorem 1.4

2.4 Examples

Part II Low Dimensional Dynamics and Thermodynamics Formalism

3 Thermodynamic Formalism and Geometric Applications for Transcendental Meromorphic and Entire Functions

3.1 Introduction

3.2 Notation

3.3 Transcendental Functions, Hyperbolicity and Expansion

3.3.1 Dynamical Preliminaries

3.3.2 Hyperbolicity and Expansion

3.3.3 Disjoint Type Entire Functions

3.4 Topological Pressure and Conformal Measures

3.4.1 Topological Pressure

3.4.2 Conformal Measures and Transfer Operator

3.4.3 Existence of Conformal Measures

3.4.4 Conformal Measures on the Radial Set and Recurrence

3.4.5 2-Conformal Measures

3.5 Perron-Frobenus-Ruelle Theorem, Spectral Gap and Applications

3.5.1 Growth Conditions

3.5.2 Geometry of Tracts

3.5.2.1 Hölder Tracts

3.5.2.2 Negative Spectrum.

3.5.2.3 Back to the Thermodynamic Formalism and Its Applications

3.6 Hyperbolic Dimension and Bowen's Formula

3.6.1 Estimates for the Hyperbolic Dimension

3.6.2 Bowen's Formula

3.7 Real Analyticity of Fractal Dimensions

3.8 Beyond Hyperbolicity

Part III Probability Theory Ergodicity and Thermodynamic Formalism

4 Recurrent Sets for Ergodic Sums of an Integer Valued Function

4.1 Introduction

4.2 Non Centered Case for d = 1

4.2.1 Preliminaries

4.2.1.1 Special Map Tf

4.2.1.2 Aperiodicity

4.2.1.3 From μ(f) &gt

0 to f ≥1

4.2.2 Sequence of Positive Density

4.2.3 Arithmetic Sequences

4.2.4 Arbitrary Sequences and Mixing Special Flows

4.2.5 Intersection of Cocycles

4.2.5.1 Return Times Theorem

4.2.6 Cocycles for T and T-1

4.3 Centered Case, d ≥1

4.3.1 Values of a Regular Cocycle

4.3.2 Hyperbolic Models

4.3.3 A Cocycle Disjoint from a Sequence with Unbounded Gaps (UGB)

4.4 Recurrent Sets for Random Walks

5 Almost Sure Invariance Principle for Random Distance Expanding Maps with a Nonuniform Decay of Correlations

5.1 Introduction

5.2 Random Distance Expanding Maps

5.2.1 Transfer Operators

5.3 A Refined Version of Gouëzel's Theorem

5.4 Main Result

6 Limit Theorem for Reflected Random Walks

6.1 Introduction and Notations

6.2 Fluctuations of Random Walks and Auxiliary Estimates

6.2.1 On the Fluctuation of Random Walks

6.2.2 Conditional Limit Theorems

6.3 On the Sub-process of Reflections

6.3.1 On the Spectrum of the Transition Probabilities Matrix R

6.3.2 A Renewal Limit Theorem for the Times of Reflections

6.4 Proof of Theorem 6.1.1

6.4.1 One-Dimensional Distribution

6.4.2 Two-Dimensional Distributions

6.4.2.1 Estimate of A1(n)

6.4.2.2 Estimate of A2 (n)

6.4.2.3 Conclusion.

6.4.3 Finite Dimensional Distributions

6.4.4 Tightness

6.5 Auxiliary Proofs

7 The Strong Borel-Cantelli Property in Conventional and Nonconventional Setups

7.1 Introduction

7.2 Preliminaries and Main Results

7.3 Proof of Theorem 7.2.2

7.3.1 The Case =1

7.3.2 The Case &gt

1

7.4 Proof of Theorem 7.2.3(i)

7.5 Proof of Theorem 7.2.3(ii)

7.6 Asymptotics of Maximums of Logarithmic Distance Functions

7.7 Asymptotics of Hitting Times

8 Application of the Convergence of the Spatio-Temporal Processes for Visits to Small Sets

8.1 Introduction

8.2 Convergence Results for Transformations and Special Flows

8.3 Number of Visits to a Small Set Before the First Visit to a Second Small Set

8.4 Number of High Records

8.5 Line Process of Random Geodesics

8.6 Time Spent by a Flow in a Small Set

Appendix: Visits by the Sinai Flow to a Finite Union of Balls in the Billiard Domain

Part IV Geometry and Thermodynamics Formation

9 Rate of Mixing for Equilibrium States in Negative Curvature and Trees

9.1 A Patterson-Sullivan Construction of Equilibrium States

9.2 Basic Ergodic Properties of Gibbs Measures

9.2.1 The Gibbs Property

9.2.2 Ergodicity

9.2.3 Mixing

9.3 Coding and Rate of Mixing for Geodesic Flows on Trees

9.3.1 Coding

9.3.2 Variational Principle for Simplicial Trees

9.3.3 Rate of Mixing for Simplicial Trees

10 Statistical Properties of the Rauzy-Veech-Zorich Map

10.1 Introduction

10.2 Interval Exchange Transformation

10.2.1 The Rauzy Class of Permutations

10.2.2 The Rauzy-Veech Renormalization T0

10.2.3 The Zorich Accelerated Renormalization T1

10.2.4 The Induced Map T2 on a Smaller Set

10.3 Transfer Operators

10.4 Statistical Properties for T2.

10.4.1 The Central Limit Theorem and Functional Central Limit Theorem

10.4.2 Almost Sure Invariance Principles

10.5 Statistical Properties for T1

10.6 Transfer Operators and Analytic Functions

10.7 Zeta Functions and Lyapunov Exponents

10.8 A Glimpse into Teichmüller Flows

10.9 Comments on Pressure

11 Entropy Rigidity, Pressure Metric, and Immersed Surfaces in Hyperbolic 3-Manifolds

11.1 Introduction

11.1.1 Outline of the Paper

11.2 Background from the Thermodynamic Formalism

11.2.1 Flows and Reparametrization

11.2.2 Periods and Measures

11.2.3 Entropy, Pressure, and Equilibrium States

11.2.4 Anosov Flows

11.2.5 A Livšic Type Theorem

11.2.6 Variance and Derivatives of the Pressure

11.2.7 The Pressure Metric

11.3 Background from Geometry

11.3.1 δ-Hyperbolic Spaces

11.3.2 Quasi-Isometries

11.3.3 Negatively Curved Manifolds and the Group of Isometries

11.3.4 Hölder Cocycles

11.3.5 Immersed Surfaces in Hyperbolic 3-Manifolds

11.3.6 Minimal Hyperbolic Germs

11.4 Immersed and Embedded Surfaces in Hyperbolic 3-Manifolds

11.4.1 Immersed Minimal Surfaces

11.4.2 Embedded Surfaces in Hyperbolic 3-Manifolds

11.4.3 The Manhattan Curve for Immersed Surfaces

11.5 Minimal Hyperbolic Germs

11.5.1 Quasifuchsian Spaces

11.5.2 Manhattan Curve for Almost-Fuchsian Space

11.5.3 Metrics on F

12 Higher Teichmüller Theory for Surface Groups and Shifts of Finite Type

12.1 Introduction

12.2 Representations and Proximality

12.3 Symbolic Dynamics

12.4 Thermodynamic Formalism

12.5 Analyticity of the Metric and the Entropy

12.6 Proof of Theorem 12.4

Part V Fractal Geometry

13 Dimension Estimates for C1 Iterated Function Systems and C1 Repellers, a Survey

13.1 Introduction

13.2 Notation.

13.2.1 Definitions of Fractal Dimensions of Sets and Measures.

Notes:

Includes bibliographical references.

ISBN:

3-030-74863-4

OCLC:

1273350946