Ergodic dynamics: from basic theory to applications / Jane Hawkins.

Format:

Book

Author/Creator:

Hawkins, Jane, 1954-

Series:

Graduate texts in mathematics ; 289.

Graduate texts in mathematics, 0072-5285 ; 289

Language:

English

Subjects (All):

Differentiable dynamical systems.

Ergodic theory.

Physical Description:

xiv, 336 pages : illustrations (black and white, and colour) ; 24 cm.

Place of Publication:

Cham : Springer International Publishing AG, [2021]

Summary:

This textbook provides a broad introduction to the fields of dynamical systems and ergodic theory. Motivated by examples throughout, the author offers readers an approachable entry-point to the dynamics of ergodic systems. Modern and classical applications complement the theory on topics ranging from financial fraud to virus dynamics, offering numerous avenues for further inquiry. Starting with several simple examples of dynamical systems, the book begins by establishing the basics of measurable dynamical systems, attractors, and the ergodic theorems. From here, chapters are modular and can be selected according to interest. Highlights include the Perron-Frobenius theorem, which is presented with proof and applications that include Google PageRank. An in-depth exploration of invariant measures includes ratio sets and type III measurable dynamical systems using the von Neumann factor classification. Topological and measure theoretic entropy are illustrated and compared in detail, with an algorithmic application of entropy used to study the papillomavirus genome. A chapter on complex dynamics introduces Julia sets and proves their ergodicity for certain maps. Cellular automata are explored as a series of case studies in one and two dimensions, including Conway's Game of Life and latent infections of HIV. Other chapters discuss mixing properties, shift spaces, and toral automorphisms. Ergodic Dynamics unifies topics across ergodic theory, topological dynamics, complex dynamics, and dynamical systems, offering an accessible introduction to the area. Readers across pure and applied mathematics will appreciate the rich illustration of the theory through examples, real-world connections, and vivid color graphics. A solid grounding in measure theory, topology, and complex analysis is assumed; appendices provide a brief review of the essentials from measure theory, functional analysis, and probability.

Contents:

Preface

The simplest examples

Dynamical Properties of Measurable Transformations

Attractors in Dynamical Systems

Ergodic Theorems

Mixing Properties of Dynamical Systems

Shift Spaces

Perron-Frobenius Theorem and Some Applications

Invariant Measures

No equivalent invariant measures: Type III maps

Dynamics of Automorphisms of the Torus and Other Groups

An Introduction to Entropy

Complex Dynamics

Maximal Entropy Measures on Julia Sets and a Computer Algorithm

Cellular Automata

Appendix A. Measures on Topological Spaces

Appendix B. Integration and Hilbert Spaces

Appendix C. Connections to Probability Theory

Bibliography

Index.

Notes:

Includes bibliographical references (pages 325-332) and index.

ISBN:

3030592413

9783030592417

OCLC:

1191214812