Dimension Theory in Dynamical Systems: Contemporary Views and Applications (Chicago Lectures in Mathematics) Review

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Dimension Theory in Dynamical Systems: Contemporary Views and Applications (Chicago Lectures in Mathematics) ReviewThis book introduces a new emerging field of inquiry and is a coherent overview of this important new subject. It presents concrete detail within a more general framework. The book introduces the cross disciplinary question: how does one use dimension theory(mathematics) to study the dynamics of a system(physics). How does one, for example, correlate the structure of an attractor seen as dimension with entropy and Lyapunov exponents? The book is current and in places exciting.
Dimension theory is related in deep ways to the dynamical parameters affecting the dimension of an invariant set. The main parameters are the characteristic contraction rates of the dynamics which have resulted in a large number of definitions of dimension (Hausdorf, box, Besicovich, correlation, and information dimensions). These definitions agree with each other when they take integer values but may not agree always.
A primary issue has been to estimate the fractal dimension of invariant sets of dynamical systems. This book provides a deep understanding of the many useful dimensional characteristics and tells how to compute these. Pesin takes his notions of dimension from one concept which unifies his approach. The book begins with generalities concerning the characteristics of dimension, sets and measures, building upon the Caratheodory construction. Pesin then applies a metric to the space to allow the Hausdorf and box dimensions to emerge as specific cases. Pesin then applies these to dimension theory within the context of dynamical systems.
Pesin then using a dynamical system approach develops a thermodynamical formalism obtaining useful dimensional definitions of topological and metric entropy and topological pressure.
The latter portion of the book uses the theory within the context of hyperbolic invariant sets and measures.
Those studying fractal invariant sets of dynamical systems will enjoy this book.Dimension Theory in Dynamical Systems: Contemporary Views and Applications (Chicago Lectures in Mathematics) Overview

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