Smooth Ergodic Theory for Endomorphisms

  • Min Qian
  • Jian-Sheng Xie
  • Shu Zhu

Part of the Lecture Notes in Mathematics book series (LNM, volume 1978)

Table of contents

  1. Front Matter
    Pages 1-11
  2. Min QUIAN, Jian-Sheng XIE, Shu ZHU
    Pages 1-8
  3. Min QUIAN, Jian-Sheng XIE, Shu ZHU
    Pages 9-13
  4. Min QUIAN, Jian-Sheng XIE, Shu ZHU
    Pages 15-26
  5. Min QUIAN, Jian-Sheng XIE, Shu ZHU
    Pages 27-44
  6. Min QUIAN, Jian-Sheng XIE, Shu ZHU
    Pages 45-86
  7. Min QUIAN, Jian-Sheng XIE, Shu ZHU
    Pages 87-96
  8. Min QUIAN, Jian-Sheng XIE, Shu ZHU
    Pages 97-150
  9. Min QUIAN, Jian-Sheng XIE, Shu ZHU
    Pages 151-171
  10. Min QUIAN, Jian-Sheng XIE, Shu ZHU
    Pages 173-204
  11. Min QUIAN, Jian-Sheng XIE, Shu ZHU
    Pages 205-244
  12. Back Matter
    Pages 1-38

About this book

Introduction

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.

Keywords

Dimension Endomorphism Ergodic Lyapunov Exponents Metric Entropy dynamical systems dynamische Systeme entropy

Authors and affiliations

  • Min Qian
    • 1
  • Jian-Sheng Xie
    • 2
  • Shu Zhu
  1. 1.Dept. MathematicsPeking UniversityBeijingChina, People's Republic
  2. 2.School of Mathematical SciencesFudan UniversityShanghaiChina, People's Republic

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-01954-8
  • Copyright Information Springer-Verlag Berlin Heidelberg 2009
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-642-01953-1
  • Online ISBN 978-3-642-01954-8
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • About this book