This book provides the first self-contained comprehensive exposition of the theory of dynamical systems as a core mathematical discipline closely intertwined with most of the main areas of mathematics. The authors introduce and rigorously develop the theory while providing researchers interested in applications with fundamental tools and paradigms.

This book provides the first self-contained comprehensive exposition of the theory of dynamical systems as a core mathematical discipline closely intertwined with most of the main areas of mathematics. The authors introduce and rigorously develop the theory while providing researchers interested in applications with fundamental tools and paradigms.

The book begins with a discussion of several elementary but fundamental examples.These are used to formulate a program for the general study of asymptotic properties and to introduce the principal theoretical concepts and methods. The main theme of the second part of the book is the interplay between local analysis near individual orbits and the global complexity of the orbit structure. The third and fourth parts develop in depth the theories of low-dimensional dynamical systems and hyperbolic dynamical systems.

The book is aimed at students and researchers in mathematics at all levels from advanced undergraduate up. Scientists and engineers working in applied dynamics, nonlinear science, and chaos will also find many fresh insights in this concrete and clear presentation. It contains more than four hundred systematic exercises.

preface

 0. introduction

1. principal branches of dynamics

2. flows, vector fields, differential equations

3. time-one map, section, suspension

4. linearization and localization

part 1 examples and fundamental concepts

 1. firstexamples

1. maps with stable asymptotic behavior

contracting maps; stability of contractions; increasing interval maps

2. linear maps

3. rotations of the circle

4. translations on the torus

5. linear flow on the torus and completely integrable systems

6. gradient flows

7. expanding maps

8. hyperbolic toral automorphisms

9. symbolic dynamical systems

sequence spaces; the shift transformation; topological markov chains; the perron-frobenius operator for positive matrices

 2. equivalence, classification, andinvariants

1. smooth conjugacy and moduli for maps equivalence and moduli; local analytic linearization; various types of moduli

2. smooth conjugacy and time change for flows

3. topological conjugacy, factors, and structural stability

4. topological classification of expanding maps on a circle expanding maps; conjugacy via coding; the fixed-point method

5. coding, horseshoes, and markov partitions

markov partitions; quadratic maps; horseshoes; coding of the toral automor- phism

6. stability of hyperbolic total automorphisms

7. the fast-converging iteration method (newton method) for the

conjugacy problem

methods for finding conjugacies; construction of the iteration process

8. the poincare-siegel theorem

9. cocycles and cohomological equations

 3. principalclassesofasymptotictopologicalinvariants

1. growth of orbits

periodic orbits and the-function; topological entropy; volume growth; topo-logical complexity: growth in the fundamental group; homological growth

2. examples of calculation of topological entropy

isometries; gradient flows; expanding maps; shifts and topological markov chains; the hyperbolic toral automorphism; finiteness of entropy of lipschitz maps; expansive maps

3. recurrence properties

 4.statistical behavior of orbits and introduction to ergodic theory

1. asymptotic distribution and statistical behavior of orbits

asymptotic distribution, invariant measures; existence of invariant measures;the birkhoff ergodic theorem; existence of symptotic distribution; ergod-icity and unique ergodicity; statistical behavior and recurrence; measure-theoretic somorphism and factors

2. examples of ergodicity; mixing

rotations; extensions of rotations; expanding maps; mixing; hyperbolic total automorphisms; symbolic systems

3. measure-theoretic entropy

entropy and conditional entropy of partitions; entropy of a measure-preserving transformation; properties of entropy

4. examples of calculation of measure-theoretic entropy

rotations and translations; expanding maps; bernoulli and markov measures;hyperbolic total automorphisms

5. the variational principle

 5.systems with smooth invar1ant measures and more examples

1. existence of smooth invariant measures

the smooth measure class; the perron-frobenius operator and divergence;criteria for existence of smooth invariant measures; absolutely continuous invariant measures for expanding maps; the moser theorem

2. examples of newtonian systems

the newton equation; free particle motion on the torus; the mathematical pendulum; central forces

3. lagrangian mechanics

uniqueness in the configuration space; the lagrange equation; lagrangian systems; geodesic flows; the legendre transform

4. examples of geodesic flows

manifolds with many symmetries; the sphere and the toms; isometrics of the hyperbolic plane; geodesics of the hyperbolic plane; compact factors; the dynamics of the geodesic flow on compact hyperbolic surfaces

5. hamiltonian systems

symplectic geometry; cotangent bundles; hamiltonian vector fields and flows;poisson brackets; integrable systems

6. contact systems

hamiltonian systems preserving a 1-form; contact forms

7. algebraic dynamics: homogeneous and afline systems

part 2 local analysis and orbit growth

 6.local hyperbolic theory and its applications

1. introduction

2. stable and unstable manifolds

hyperbolic periodic orbits; exponential splitting; the hadamard-perron the-orem; proof of the hadamard-perron theorem; the inclination lemma

3. local stability of a hyperbolic periodic point

the hartman-grobman theorem; local structural stability

4. hyperbolic sets

definition and invariant cones; stable and unstable manifolds; closing lemma and periodic orbits; locally maximal hyperbolic sets

5. homoclinic points and horseshoes

general horseshoes; homoclinic points; horseshoes near homoclinic poi

6. local smooth linearization and normal forms

jets, formal power series, and smooth equivalence; general formal analysis; the hyperbolic smooth case

 7.transversality and genericity

1. generic properties of dynamical systems

residual sets and sets of first category; hyperbolicity and genericity

2. genericity of systems with hyperbolic periodic points

transverse fixed points; the kupka-smale theorem

3. nontransversality and bifurcations

structurally stable bifurcations; hopf bifurcations

4. the theorem of artin and mazur

 8.orbitgrowtharisingfromtopology

1. topological and fundamental-group entropies

2. a survey of degree theory

motivation; the degree of circle maps; two definitions of degree for smooth maps; the topological definition of degree

3. degree and topological entropy

4. index theory for an isolated fixed point

5. the role of smoothness: the shub-sullivan theorem

6. the lefschetz fixed-point formula and applications

7. nielsen theory and periodic points for toral maps

 9.variational aspects of dynamics

1. critical points of functions, morse theory, and dynamics

2. the billiard problem

3. twist maps

definition and examples; the generating function; extensions; birkhoff peri-odic orbits; global minimality of birkhoff periodic orbits

4. variational description of lagrangian systems

5. local theory and the exponential map

6. minimal geodesics

7. minimal geodesics on compact surfaces

part 3 low-dimensional phenomena

 10. introduction: what is low-dimensional dynamics?

motivation; the intermediate value property and conformality; vet low-dimensional and low-dimensional systems; areas of !ow-dimensional dynamics

 11.homeomorphismsofthecircle

1. rotation number

2. the poincare classification

rational rotation number; irrational rotation number; orbit types and mea-surable classification

 12. circle diffeomorphisms

1. the denjoy theorem

2. the denjoy example

3. local analytic conjugacies for diophantine rotation number

4. invariant measures and regularity of conjugacies

5. an example with singular conjugacy

6. fast-approximation methods

conjugacies of intermediate regularity; smooth cocycles with wild cobound-aries

7. ergodicity with respect to lebesgue measure

 13. twist maps

1. the regularity lemma

2. existence of aubry-mather sets and homoclinic orbits

aubry-mather sets; invariant circles and regions of instability

3. action functionals, minimal and ordered orbits

minimal action; minimal orbits; average action and minimal measures; stable sets for aubry-mather sets

4. orbits homoclinic to aubry-mather sets

5. nonexisience of invariant circles and localization of aubry-mather sets

 14.flowsonsurfacesandrelateddynamicalsystems

1. poincare-bendixson theory

the poincare-bendixson theorem; existence of transversals

2. fixed-point-free flows on the torus

global transversals; area-preserving flows

3. minimal sets

4. new phenomena

the cherry flow; linear flow on the octagon

5. interval exchange transformations

definitions and rigid intervals; coding; structure of orbit closures; invariant measures; minimal nonuniquely ergodic interval exchanges

6. application to flows and billiards

classification of orbits; parallel flows and billiards in polygons

7. generalizations of rotation number

rotation vectors for flows on the torus; asymptotic cycles; fundamental class and smooth classification of area-preserving flows

 15.continuousmapsoftheinterval

1. markov covers and partitions

2. entropy, periodic orbits, and horseshoes

3. the sharkovsky theorem

4. maps with zero topological entropy

5. the kneading theory

6. the tent model

 16.smoothmapsoftheinterval

1. the structure of hyperbolic repellers

2. hyperbolic sets for smooth maps

3. continuity of entropy

4. full families of unimodal maps

part 4 hyperbolic dynamical systems

 17.surveyofexamples

1. the smale attractor

2. the da (derived from anosov) map and the plykin attractor

the da map; the plykln attractor

3. expanding maps and anosov automorphisms of nilmanifolds

4. definitions and basic properties of hyperbolic sets for flows

5. geodesic flows on surfaces of constant negative curvature

6. geodesic flows on compact riemannian manifolds with negative sectional curvature

7. geodesic flows on rank-one symmetric spaces

8. hyperbolic julia sets in the complex plane

rational maps of the riemann sphere; holomorphic dynamics

 18.topologicalpropertiesofhyperbolicsets

1. shadowing of pseudo-orbits

2. stability of hyperbolic sets and markov approximation

3. spectral decomposition and specification

spectral decomposition for maps; spectral decomposition for flows; specifica- tion

4. local product structure

5. density and growth of periodic orbits

6. global classification of anosov diffeomorphisms on tori

7. markov partitions

 19. metric structure of hyperbolic sets

1. holder structures

the invariant class of hsider-continuons functions; hslder continuity of conju-gacies; hslder continuity of orbit equivalence for flows; hslder continuity and differentiability of the unstable distribution; hslder continuity of the jacobian

2. cohomological equations over hyperbolic dynamical systems

the livschitz theorem; smooth invariant measures for anosov diffeomor-phisms; time change and orbit equivalence for hyperbolic flows; equivalence of torus extensions

 20.equilibriumstatesandsmoothinvariantmeasures

1. bowen measure

2. pressure and the variational principle

3. uniqueness and classification of equilibrium states

uniqueness of equilibrium states; classification of equilibrium states

4. smooth invariant measures

properties of smooth invariant measures; smooth classification of anosov dif-feomorphisms on the torus; smooth classification of contact anosov flows on 3-manifolds

5. margulis measure

6. multiplicative asymptotic for growth of periodic points

local product flow boxes; the multiplicative asymptotic of orbit growth

supplement

 s. dynamical systems with nonuniformly hyperbolic behavior byanatolekatokandleonardomendoza

1. introduction

2. lyapunov exponents

cocycles over dynamical systems; examples of cocycles; the multiplicative ergodic theorem; osedelec-pesin e-reduction theorem; the rue!!e inequality

3. regular neighborhoods

existence of regular neighborhoods; hyperbolic points, admissible manifolds, and the graph transform

4. hyperbolic measures

preliminaries; the closing lemma; the shadowing lemma; pseudo-markov covers; the livschitz theorem

5. entropy and dynamics of hyperbolic measures

hyperbolic measures and hyperbolic periodic points; continuous measures and transverse homoclinic points; the spectral decomposition theorem; entropy,horseshoes, and periodic points for hyperbolic measures

appendix

 a. background material

1. basic topology

topological spaces; homotopy theory; metric spaces

2. functional analysis

3. differentiable manifolds

differentiable manifolds; tensor bundles; exterior calculus; transversality

4. differential geometry

5. topology and geometry of surfaces

6. measure theory

basic notions; measure and topology

7. homology theory

8. locally compact groups and lie groups

notes

hintsandanswerstotheexercises

references

index