This monograph offers a coherent, self-contained account of the theory of Sinai–Ruelle–Bowen measures and decay of correlations for nonuniformly hyperbolic dynamical systems. A central topic in the statistical theory of dynamical systems, the book in particular provides a detailed exposition of the theory developed by L.-S. Young for systems admitting induced maps with certain analytic and geometric properties. After a brief introduction and preliminary results, Chapters 3, 4, 6 and 7 provide essentially the same pattern of results in increasingly interesting and complicated settings. Each chapter builds on the previous one, apart from Chapter 5 which presents a general abstract framework to bridge the more classical expanding and hyperbolic systems explored in Chapters 3 and 4 with the nonuniformly expanding and partially hyperbolic systems described in Chapters 6 and 7. Throughout the book, the theory is illustrated with applications. A clear and detailed account of topics of current research interest, this monograph will be of interest to researchers in dynamical systems and ergodic theory. In particular, beginning researchers and graduate students will appreciate the accessible, self-contained presentation.
Author(s): José F. Alves
Series: Springer Monographs in Mathematics
Publisher: Springer
Year: 2021
Language: English
Pages: 259
City: Cham
Preface
Contents
1 Introduction
1.1 Physical Measures
1.2 SRB Measures
1.3 Decay of Correlations
References
2 Preliminaries
2.1 Partitions
2.1.1 Generating Partitions
2.1.2 Bases
2.1.3 Measurable Partitions
2.2 Jacobians
2.3 Basins
References
3 Expanding Structures
3.1 Gibbs-Markov Maps
3.1.1 Bounded Distortion
3.1.2 A Space for the Densities
3.1.3 Equilibrium Measures
3.2 Induced Maps
3.3 Tower Maps
3.3.1 Tower Extension
3.3.2 Convergence to the Equilibrium
3.3.3 Decay of Correlations
3.4 Lifting Observables
3.5 Application: Intermittent Maps
3.5.1 Neutral Fixed Point
3.5.2 Interval Map
3.5.3 Circle Map
References
4 Hyperbolic Structures
4.1 Young Structures
4.1.1 Quotient Return Map
4.1.2 Bounded Distortion
4.2 SRB Measures
4.2.1 Return Map
4.2.2 Original Dynamics
4.3 Tower Extension
4.3.1 Quotient Tower
4.4 Decay of Correlations
4.4.1 Reducing to the Quotient Tower
4.4.2 Regularity of the Discretisation
4.4.3 Specific Rates
4.4.4 The Non-exact Case
4.5 Regularity of the Stable Holonomy
4.5.1 Absolute Continuity
4.5.2 The Density Formula
4.6 Application: A Solenoid with Intermittency
4.6.1 Partially Hyperbolicity
4.6.2 Positive Lyapunov Exponent
4.6.3 Young Structure
References
5 Inducing Schemes
5.1 A General Framework
5.1.1 Bounded Distortion
5.2 The Partition
5.2.1 Inductive Construction
5.2.2 Key Relations
5.2.3 Metric Estimates
5.3 Inducing Times
5.3.1 Integrability
5.3.2 Tail Estimates
References
6 Nonuniformly Expanding Attractors
6.1 Nonuniform Expansion and Slow Recurrence
6.1.1 Hyperbolic Times and Preballs
6.2 Attractors
6.2.1 Ergodic Components
6.2.2 Unshrinkable Sets
6.3 Gibbs-Markov Induced Maps
6.4 SRB Measures
6.5 Decay of Correlations
6.6 Applications
6.6.1 Derived from Expanding
6.6.2 Viana Maps
References
7 Partially Hyperbolic Attractors
7.1 Dominated Splitting
7.1.1 Hölder Control of Centre-Unstable Disks
7.1.2 Hyperbolic Times and Predisks
7.1.3 Partial Hyperbolicity
7.2 Attractors
7.2.1 Ergodic Components
7.2.2 Unshrinkable Sets
7.3 Young Structure
7.3.1 Inducing Domain
7.3.2 Product Structure
7.3.3 Recurrence Times
7.4 SRB Measures
7.5 Decay of Correlations
7.6 Application: Derived from Anosov
References
Index
