The origins of dynamical systems trace back to flows and differential equations, and this is a modern text and reference on dynamical systems in which continuous-time dynamics is primary. It addresses needs unmet by modern books on dynamical systems, which largely focus on discrete time. Students have lacked a useful introduction to flows, and researchers have difficulty finding references to cite for core results in the theory of flows. Even when these are known substantial diligence and consultation with experts is often needed to find them.
This book presents the theory of flows from the topological, smooth, and measurable points of view. The first part introduces the general topological and ergodic theory of flows, and the second part presents the core theory of hyperbolic flows as well as a range of recent developments. Therefore, the book can be used both as a textbook – for either courses or self-study – and as a reference for students and researchers.
There are a number of new results in the book, and many more are hard to locate elsewhere, often having appeared only in the original research literature. This book makes them all easily accessible and does so in the context of a comprehensive and coherent presentation of the theory of hyperbolic flows.
Keywords: hyperbolic, hyperbolicity, flow, ergodic theory, topological dynamics, rigidity, expansiveness, shadowing, specification, geodesic flow, Anosov flow, Axiom A, entropy, equilibrium states, stable manifold, topological pressure, symbolic flows, Markov partitions
Author(s): Todd Fisher, Boris Hasselblatt
Series: Zurich Lectures in Advanced Mathematics
Publisher: European Mathematical Society
Year: 2019
Acknowledgments
Introduction
About this book
Continuous and discrete time
Historical sketch
I Flows
Topological dynamics
Basic properties
Time change, flow under a function, and sections
Conjugacy and orbit equivalence
Attractors and repellers
Recurrence properties and chain decomposition
Transitivity, minimality, and topological mixing
Expansive flows
Weakening expansivity*
Symbolic flows, coding
Hyperbolic geodesic flow*
Isometries, geodesics, and horocycles of the hyperbolic plane and disk
Dynamics of the natural flows
Compact factors
The geodesic flow on compact hyperbolic surfaces
Symmetric spaces
Hamiltonian systems
Ergodic theory
Flow-invariant measures and measure-preserving transformations
Ergodic theorems
Ergodicity
Mixing
Invariant measures under time change
Flows under a function
Spectral theory*
Entropy, pressure, and equilibrium states
Measure-theoretic entropy
Topological entropy
Topological pressure and equilibrium states
Equilibrium states for time-t maps*
II Hyperbolic flows
Introduction to Part II
Hyperbolicity
Hyperbolic sets and basic properties
Physical flows: Geodesic flows, magnetic flows, billiards, gases, and linkages
Shadowing, expansivity, closing, specification, and Axiom A
The Anosov Shadowing Theorem, structural and \Omega-stability
Local linearization: The Hartman–Grobman Theorem
The Mather–Moser method*
Invariant foliations
Stable and unstable foliations
Global foliations, local maximality, Bowen bracket
Livshitz theory
Hölder continuity of orbit equivalence
Horseshoes and attractors
Markov partitions
Failure of local maximality*
Smooth linearization and normal forms*
Differentiability in the Hartman–Grobman Theorem*
Ergodic theory of hyperbolic sets
The Hopf argument, absolute continuity, mixing
Stable ergodicity*
Specification, uniqueness of equilibrium states
Sinai–Ruelle–Bowen measures
Hamenstädt–Margulis measure*
Asymptotic orbit growth*
Rates of mixing*
Anosov flows
Anosov diffeomorphisms, suspensions, and mixing
Foulon–Handel–Thurston surgery
Anomalous Anosov flows
Codimension-1 Anosov flows
\mathbb{R}-covered Anosov 3-flows
Horocycle and unstable flows*
Rigidity
Multidimensional time: Commuting flows
Conjugacies
Entropy and Lyapunov exponents
Optimal regularity of the invariant subbundles
Longitudinal regularity
Sharpness for transversely symplectic flows, threading
Smooth invariant foliations
Godbillon–Vey invariants*
Measure-theoretic entropy of maps
Lebesgue spaces
Entropy and conditional entropy
Properties of entropy
Hyperbolic maps and invariant manifolds
The Contraction Mapping Principle
Generalized eigenspaces
The spectrum of a linear map
Hyperbolic linear maps
Admissible manifolds: The Hadamard method
The Inclination Lemma and homoclinic tangles
Absolute continuity
Hints and answers to the exercises
Bibliography
Index of persons
Index
Index of theorems
