| Description |
1 online resource (xiii, 277 pages) : illustrations |
| Series |
Lecture notes in mathematics, 0075-8434 ; 1978 |
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Lecture notes in mathematics (Springer-Verlag) ; 1978.
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| Contents |
Cover -- Contents -- I Preliminaries -- I.1 Metric Entropy -- I.2 Multiplicative Ergodic Theorem -- I.3 Inverse Limit Space -- II Margulis-Ruelle Inequality -- II. 1 Statement of the Theorem -- II. 2 Preliminaries -- II. 3 Proof of the Theorem -- III Expanding Maps -- III. 1 Main Results -- III. 2 Proof of Theorem III. 1.1 -- III. 3 Basic Facts About Expanding Maps -- III. 4 Proofs of Theorems III. 1.2 and III. 1.3 -- IV Axiom A Endomorphisms -- IV. 1 Introduction and Main Results -- IV. 2 Preliminaries -- IV. 3 Volume Lemma and the H246;lder Continuity of 966;u -- IV. 4 Equilibrium States of 966;u on 923;f -- IV. 5 Pesin8217;s Entropy Formula -- IV. 6 Large Ergodic Theorem and Proof of Main Theorems -- V Unstable and Stable Manifolds for Endomorphisms -- V.1 Preliminary Facts -- V.2 Fundamental Lemmas -- V.3 Some Technical Facts About Contracting Maps -- V.4 Local Unstable Manifolds -- V.5 Global Unstable Sets -- V.6 Local and Global Stable Manifolds -- V.7 H246;lder Continuity of Sub-bundles -- V.8 Absolute Continuity of Families of Submanifolds -- V.9 Absolute Continuity of Conditional Measures -- VI Pesin8217;s Entropy Formula for Endomorphisms -- VI. 1 Main Results -- VI. 2 Preliminaries -- VI. 3 Proof of Theorem VI. 1.1 -- VII SRB Measures and Pesin8217;s Entropy Formula for Endomorphisms -- VII. 1 Formulation of the SRB Property and Main Results -- VII. 2 Technical Preparations for the Proof of the Main Result -- VII. 3 Proof of the Sufficiency for the Entropy Formula -- VII. 4 Lyapunov Charts -- VII. 5 Local Unstable Manifolds and Center Unstable Sets -- VII. 5.1 Local Unstable Manifolds and Center Unstable Sets -- VII. 5.2 Some Estimates -- VII. 5.3 Lipschitz Property of Unstable Subspaces within Center Unstable Sets -- VII. 6 Related Measurable Partitions -- VII. 6.1 Partitions Adapted to Lyapunov Charts -- VII. 6.2 More on Increasing Partitions -- VII. 6.3 Two Useful Partitions -- VII. 6.4 Quotient Structure -- VII. 6.5 Transverse Metrics -- VII. 7 Some Consequences of Besicovitch8217;s Covering Theorem -- VII. 8 The Main Proposition -- VII. 9 Proof of the Necessity for the Entropy Formula -- VII. 9.1 The Ergodic Case -- VII. 9.2 The General Case -- VIII Ergodic Property of Lyapunov Exponents -- VIII. 1 Introduction and Main Results -- VIII. 2 Lyapunov Exponents of Axiom A Attractors of Endomorphisms -- VIII. 3 Nonuniformly Completely Hyperbolic Attractors -- IX Generalized Entropy Formula -- IX. 1 Related Notions and Statements of the Main Results -- IX. 1.1 Pointwise Dimensions and Transverse Dimensions -- IX. 1.2 Statements of the Main Results -- IX. 2 Preliminaries -- IX. 2.1 Some Estimations on Unstable Manifolds -- IX. 2.2 Related Partitions -- IX. 2.3 Transverse Metrics on i(x)/- with 2iu -- IX. 2.4 Entropies of the Related Partitions -- IX. 3 Definitions of Local Entropies along Unstable Manifolds -- IX. 4 Estimates of Local Entropies along Unstable Manifolds -- IX. 4.1 Estimate of Local Entropy h -- IX. 4.2 Estimate of Local Entropy hi from Below with 2iu -- IX. 4.3 Estimate of Local Entropy hi from Above with 2iu -- IX. 5 The General Case: without Ergodic Assumption -- X Exact Dimensionality of Hyperbolic Measures -- X.1 Expanding Maps8217; Case8211;Proof of Theorem X.0.1 -- X.2 Diffeomorphisms8217; Case8211;Proof |
| Summary |
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 |
| Bibliography |
Includes bibliographical references (pages 271-274) and index |
| Notes |
Print version record |
| Subject |
Ergodic theory.
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Endomorphisms (Group theory)
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Endomorphisms (Group theory)
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Ergodic theory
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Ergodentheorie
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Dynamisches System
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Endomorphismus
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| Form |
Electronic book
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| Author |
Xie, Jian-sheng.
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Zhu, Shu, 1964-
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| ISBN |
9783642019548 |
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3642019544 |
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9783642019531 |
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3642019536 |
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1282655809 |
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9781282655805 |
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