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Typical dynamics of volume preserving homeomorphisms 1st Edition by Steve Alpern ISBN 0521582873 9780521582872

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Typical dynamics of volume preserving homeomorphisms 1st Edition Steve Alpern Digital Instant Download

Author(s): Steve Alpern, V. S. Prasad
ISBN(s): 9780521582872, 0521582873
Edition: 1
File Details: PDF, 1.21 MB
Year: 2001
Language: english
SKU: EB-1014694 Category: Tags: ,

Typical dynamics of volume preserving homeomorphisms 1st Edition by Steve Alpern – Ebook PDF Instant Download/Delivery: 0521582873, 9780521582872
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Product details:

ISBN 10: 0521582873 
ISBN 13: 9780521582872
Author: Steve Alpern

This 2000 book provides a self-contained introduction to typical properties of homeomorphisms. Examples of properties of homeomorphisms considered include transitivity, chaos and ergodicity. A key idea here is the interrelation between typical properties of volume preserving homeomorphisms and typical properties of volume preserving bijections of the underlying measure space. The authors make the first part of this book very concrete by considering volume preserving homeomorphisms of the unit n-dimensional cube, and they go on to prove fixed point theorems (Conley–Zehnder– Franks). This is done in a number of short self-contained chapters which would be suitable for an undergraduate analysis seminar or a graduate lecture course. Much of this work describes the work of the two authors, over the last twenty years, in extending to different settings and properties, the celebrated result of Oxtoby and Ulam that for volume homeomorphisms of the unit cube, ergodicity is a typical property.

Typical dynamics of volume preserving homeomorphisms 1st Table of contents:

  • Introduction

    • Overview of volume preserving homeomorphisms

    • Historical background and motivation

    • Main results and structure of the book

  • Preliminaries

    • Topological and measure-theoretic concepts

    • Basic definitions: homeomorphisms, volume preservation

    • Ergodic theory fundamentals

  • Volume Preserving Transformations

    • Properties of volume preserving maps

    • Examples and constructions

    • Invariant measures and ergodicity

  • Generic Properties in Dynamical Systems

    • Baire category approach

    • Typicality and genericity in dynamics

    • Residual sets and generic properties

  • Mixing and Ergodic Properties

    • Weak mixing, strong mixing

    • Bernoulli systems

    • Spectral theory of volume preserving homeomorphisms

  • Structural Stability and Rigidity

    • Stability concepts in volume preserving dynamics

    • Structural rigidity phenomena

    • Perturbation techniques

  • Chaos and Complexity

    • Topological entropy

    • Lyapunov exponents

    • Horseshoes and symbolic dynamics

  • Constructions of Typical Dynamics

    • Techniques for constructing typical volume preserving maps

    • Approximation methods

    • Examples and applications

  • Applications and Further Directions

    • Connections to fluid dynamics, Hamiltonian systems

    • Open problems and conjectures

    • Future research perspectives

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Tags: Steve Alpern, Typical dynamics, homeomorphisms