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Introduction to the Modern Theory of Dynamical Systems : Encyclopedia of Mathematics and Its Applications - Anatole Katok

Introduction to the Modern Theory of Dynamical Systems

Encyclopedia of Mathematics and Its Applications

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Published: 9th February 2011
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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 the theories of low-dimensional dynamical systems and hyperbolic dynamical systems in depth. Over 400 systematic exercises are included in the text. 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.

' ... there is no other treatment coming close in terms of comprehensiveness and readability ... it is indispensable for anybody working on dynamical systems in almost any context, and even experts will find interesting new proofs, insights and historical references throughout the book.' Monatshefte fur Mathematik '... contains detailed discussion ... presents many recent results ... The text is carefully written and is accompanied by many excercises.' European Mathematical Society Newsletter 'This book provides the first self-contained comprehensive exposition of the theory of dynamical systems as a core mathematical discipline.' L'Enseignement Mathematique

Examples and Fundamental Concepts
Introduction
First examples
Equivalence, classification, and invariants
Principle classes of asymptotic invariants
Statistical behavior of the orbits and introduction to ergodic theory
Smooth invariant measures and more examples
Local Analysis and Orbit Growth
Local hyperbolic theory and its applications
Transversality and genericity
Orbit growth arising from topology
Variational aspects of dynamics
Low-Dimensional Phenomena
Introduction: What is low dimensional dynamics
Homeomorphisms of the circle
Circle diffeomorphisms
Twist maps
Flows on surfaces and related dynamical systems
Continuous maps of the interval
Smooth maps of the interval
Hyperbolic Dynamical Systems
Survey of examples
Topological properties of hyperbolic sets
Metric structure of hyperbolic sets
Equilibrium states and smooth invariant measures
Sopplement and Appendix
Dynamical systems with nonuniformly hyperbolic behavior
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780521341875
ISBN-10: 0521341876
Series: Encyclopedia of Mathematics and Its Applications
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 824
Published: 9th February 2011
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 23.4 x 15.6  x 5.9
Weight (kg): 1.27