Smooth Ergodic Theory for Endomorphisms [Paperback]

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.

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.

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. (M?rio Bessa, Mathematical Reviews, Issue 2010 m)

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 Pesins 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.lÓ'