Computational Ergodic Theory

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Springer Science & Business Media, Feb 11, 2005 - Mathematics - 453 pages
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Ergodic theory is hard to study because it is based on measure theory, which is a technically difficult subject to master for ordinary students, especially for physics majors. Many of the examples are introduced from a different perspective than in other books and theoretical ideas can be gradually absorbed while doing computer experiments. Theoretically less prepared students can appreciate the deep theorems by doing various simulations. The computer experiments are simple but they have close ties with theoretical implications. Even the researchers in the field can benefit by checking their conjectures, which might have been regarded as unrealistic to be programmed easily, against numerical output using some of the ideas in the book. One last remark: The last chapter explains the relation between entropy and data compression, which belongs to information theory and not to ergodic theory. It will help students to gain an understanding of the digital technology that has shaped the modern information society.

 

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Contents

tents
1
Z Measures and Lebesgue Integration
7
Compact Abelian Groups and Characters
14
Continued Fractions
20
variant Measures
47
Isomorphic Transformations
61
he Birkhoff Ergodic Theorem
85
Gelfands Problem
99
Z Number of Significant Digits and the Divergence Speed
279
Speed of Approximation by Convergents
287
Multidimensional Case
299
The Lyapunov Exponent of a Differentiable Mapping
311
able and Unstable Manifolds
333
The Standard Mapping
339
Maple Programs
345
ecurrence and Entropy
363

Maple Programs
113
he Central Limit Theorem
133
Speed of Correlation Decay
143
Iore on Ergodicity
155
How to Sketch a Conjugacy Using Rotation Number
195
Uniform Distribution
211
Mod 2 Normality Conditions
219
How to Sketch a Cobounding Function
227
148
241
OneDimensional Case
270
153
276
The Nonoverlapping First Return Time
369
Product of the Return Time and the Probability
376
ecurrence and Dimension
391
Singular Continuous Invariant Measures
398
Maple Programs
409
ata Compression
417
Huffman Coding
423
Maple Programs
429
ances
439
156
450
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