Aspects of Ergodic, Qualitative and Statistical Theory of Motion

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Springer Science & Business Media, Mar 23, 2004 - Mathematics - 440 pages
Intended for beginners in ergodic theory, this book addresses students as well as researchers in mathematical physics. The main novelty is the systematic treatment of characteristic problems in ergodic theory by a unified method in terms of convergent power series and renormalization group methods, in particular. Basic concepts of ergodicity, like Gibbs states, are developed and applied to, e.g., Asonov systems or KAM Theory. Many examples illustrate the ideas and, in addition, a substantial number of interesting topics are treated in the form of guided problems.
 

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Contents

General Qualitative Properties
1
12 Examples and Some Definitions
3
13 Harmonic Oscillators and Integrable Systems as Dynamical Systems
13
14 Frequencies of Visit
17
Ergodicity and Ergodic Points
27
22 Ergodic Properties of QuasiPeriodic Motions Ergodic Sequences and Measures
32
23 Ergodic Points
52
24 Ergodic Decomposition
62
72 Cluster Expansions
240
73 Renormalization by Decimation in OneDimensional Systems
257
More Criteria
270
75 Phase Transitions
276
Special Ergodic Theory Problems in Nonchaotic Dynamics
291
82 Graphs and Diagrams for the Lindstedt Series
299
83 Cancellations
307
84 Convergence and the KAM Theorem
316

Entropy and Complexity
73
32 The ShannonMcMillan Theorem
81
33 Elementary Properties of the Average Entropy
92
34 Further Properties of the Average Entropy Generator Theorem
99
Markovian Pavements
109
42 Markovian Pavements for Hyperbolic Systems
119
43 Coding of the Volume Measure of Smooth Hyperbolic Systems
141
Gibbs Distributions
155
52 Properties of Gibbs Distributions
165
53 Gibbs Distributions on Z+
170
Expansive Maps of 01
180
General Properties of Gibbs and SRB Distributions
189
62 Applications to Anosov Systems SRB Distribution
199
63 Periodic Orbits Invariant Probability Distributions and Entropy
208
64 Equivalent Potentials Gibbs Distributions with Transitive Vacuum
216
Analyticity Singularity and Phase Transitions
227
Some Special Topics in KAM Theory
327
92 Bounds on Renormalized Series Convergence
332
93 Scaling Laws for the Standard Map
341
94 Limit Function for the Standard Map near Resonances
352
Special Problems in Chaotic Dynamics
359
102 Extended Systems Lattices of Arnolds Cat Maps
366
An SRB Distribution
371
104 Chaos in SpaceTime and SRB Distributions
381
105 Isomorphisms
392
A Nonequilibrium Thermodynamics? TwentySeven Comments
397
Bibliography
411
Name Index
423
Subject Index
425
Citations Index
433
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