Smooth Ergodic Theory for Endomorphisms
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 c
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1978
Min Qian · Jian-Sheng Xie · Shu Zhu
Smooth Ergodic Theory for Endomorphisms
123
Editors Min QIAN School of Mathematical Sciences Peking University Beijing 100871 P. R. China
Shu ZHU 6807 Edenwood Drive Mississauga, ON L5N 3X9 Canada [email protected]
Jian-Sheng XIE School of Mathematical Sciences Fudan University Shanghai 200433 P. R. China [email protected]
ISBN: 978-3-642-01953-1 DOI: 10.1007/978-3-642-01954-8
e-ISBN: 978-3-642-01954-8
Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2009928105 Mathematics Subject Classification (2000): 37C40, 37C45, 37D20, 37H15 c Springer-Verlag Berlin Heidelberg 2009 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publisher Services Printed on acid-free paper springer.com
Comme souvenir de mon amiti´e avec le Professeur Y. -S. Sun, je voudrais mentionner que c’est lui qui m’a fait connaˆıtre la formule de Pesin pour la premi`ere fois de ma vie. Min Qian
Preface
Smooth ergodic theory of deterministic dynamical systems deals with the study of dynamical behaviors relevant to certain invariant measures under differentiable mappings or flows. The relevance of invariant measures is that they describe the frequencies of visits for an orbit and hence they give a probabilistic description of the evolution of a dynamical system. The fact that the system is differentiable allows one to use techniques from analysis and geometry. The study of transformations and their long-term behavior is ubiquitous in mathematics and the sciences. They arise not only in applications to the real world but also to diverse mathematical disciplines, including number theory, Lie groups, algorithms, Riemannian geometry, etc. Hence smooth ergodic theory is the meeting ground of many different ideas in pure and applied mathematics. It has witnessed a great progress since the pioneering works of Sinai, Ruelle and Bowen on Axiom A diffeomorphisms and of Pesin on non-uniformly hyperbolic systems, and now it becomes a well-developed field. In this theory, among the major concepts are the notions of Lyapunov exponents and metric entropy. Lyapunov exponent describes the exponential rate of expansion or cont
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