this book provides the first self-contained comprehensiveexposition of the theory of dynamical systems as a coremathematical discipline closely intertwined with most of the mainareas of mathematics. the authors introduce and rigorously developthe theory while providing researchers interested in applicationswith fundamental tools and paradigms.
the book begins with a discussion of several elementary butfundamental examples. these are used to formulate a program for thegeneral study of asymptotic properties and to introduce theprincipal theoretical concepts and methods. the main theme of thesecond part of the book is the interplay between local analysisnear individual orbits and the global complexity of the orbitstructure. the third and fourth parts develop in depth the theoriesof !ow-dimensional dynamical systems and hyperbolic dynamicalsystems.
the book is aimed at students and researchers in mathematics atall levels from ad-vanced undergraduate up. scientists andengineers working in applied dynamics， non-linear science， andchaos will also find many fresh insights in this concrete and clearpresentation. it contains more than four hundred systematicexercises.
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 1examples 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
r
编者：(美国)卡托克 (Katok A.)
