Smooth Ergodic Theory for Endomorphisms

ISBN-13: 9783642019531 / Angielski / Miękka / 2009 / 300 str.

Quian Min; Jian-Sheng Xie; Shu Zhu

Opis

Ideal for researchers and graduate students, this volume sets out a general smooth ergodic theory for deterministic dynamical systems generated by non-invertible endomorphisms. Its focus is on the relations between entropy, Lyapunov exponents and dimensions.

Recenzje

From the reviews:

"In the useful monograph under review the authors intend to assemble several topics in the classic ergodic theory of deterministic endomorphisms gathering the most important results available until the present time. ... should be of interest to mathematicians, postgraduate students and physicists working on this field." (Mário Bessa, Mathematical Reviews, Issue 2010 m)

Spis treści

Preliminaries.- Margulis-Ruelle Inequality.- Expanding Maps.- Axiom A Endomorphisms.- Unstable and Stable Manifolds for Endomorphisms.- Pesin#x2019;s Entropy Formula for Endomorphisms.- SRB Measures and Pesin#x2019;s Entropy Formula for Endomorphisms.- Ergodic Property of Lyapunov Exponents.- Generalized Entropy Formula.- Exact Dimensionality of Hyperbolic Measures.

Blurb

This volume presents a general smooth ergodic theory for deterministic dynamical systems generated by non-invertible endomorphisms, mainly concerning the relations between entropy, Lyapunov exponents and dimensions.
The authors make extensive use of the combination of the inverse limit space technique and the techniques developed to tackle random dynamical systems. The most interesting results in this book are (1) the equivalence between the SRB property and Pesin’s entropy formula; (2) the generalized Ledrappier-Young entropy formula; (3) exact-dimensionality for weakly hyperbolic diffeomorphisms and for expanding maps. The proof of the exact-dimensionality for weakly hyperbolic diffeomorphisms seems more accessible than that of Barreira et al. It also inspires the authors to argue to what extent the famous Eckmann-Ruelle conjecture and many other classical results for diffeomorphisms and for flows hold true.
After a careful reading of the book, one can systematically learn the Pesin theory for endomorphisms as well as the typical tricks played in the estimation of the number of balls of certain properties, which are extensively used in Chapters IX and X.