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Smooth ergodic theory for endomorphisms

Author: Min Qian; Jian-sheng Xie; Shu Zhu
Publisher: Berlin : Springer, ©2009.
Series: Lecture notes in mathematics (Springer-Verlag), 1978.
Edition/Format:   eBook : Document : EnglishView all editions and formats
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)  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Qian, Min.
Smooth ergodic theory for endomorphisms.
Berlin : Springer, ©2009
(DLC) 2009928105
(OCoLC)401153872
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Min Qian; Jian-sheng Xie; Shu Zhu
ISBN: 9783642019548 3642019544 9783642019531 3642019536 1282655809 9781282655805
OCLC Number: 656399529
Description: 1 online resource (xiii, 277 pages) : illustrations.
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.
Series Title: Lecture notes in mathematics (Springer-Verlag), 1978.
Responsibility: Min Qian, Jian-Sheng Xie, Shu Zhu.
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Abstract:

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  Read more...

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Primary Entity

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   schema:description "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."@en ;
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