Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- PART I Entropy in ergodic theory
- PART II Entropy in topological dynamics
- 6 Topological entropy
- 7 Dynamics in dimension zero
- 8 The entropy structure
- 9 Symbolic extensions
- 10 A touch of smooth dynamics*
- PART III Entropy theory for operators
- Appendix A Toolbox
- Appendix B Conditional S–M–B
- List of symbols
- References
- Index
8 - The entropy structure
from PART II - Entropy in topological dynamics
Published online by Cambridge University Press: 07 October 2011
- Frontmatter
- Contents
- Preface
- Introduction
- PART I Entropy in ergodic theory
- PART II Entropy in topological dynamics
- 6 Topological entropy
- 7 Dynamics in dimension zero
- 8 The entropy structure
- 9 Symbolic extensions
- 10 A touch of smooth dynamics*
- PART III Entropy theory for operators
- Appendix A Toolbox
- Appendix B Conditional S–M–B
- List of symbols
- References
- Index
Summary
The topological entropy of a dynamical system is a rather crude measurement of its complexity. In order to thoroughly understand the dynamics it is essential not only to replace the topological entropy by the entropy function µ ↦ h(µ, T) on invariant measures, but also to replace the topological entropy detectable in a given resolution, say h1(T, ε) or h2(T, ε), by some function, say µ ↦ h(µ, T, ε), reflecting the dynamical entropy of each measure detectable in that resolution. Such a function has not been presented in this book yet. As we shall see, there is an essential difference between entropy of a measure with respect to a measurable resolution, i.e., a partition (even if we require that the cells have diameters bounded by ε) and the entropy of a measure with respect to a topological resolution, which we are about to define. The difference is in semicontinuity properties and in the “type of convergence” to the entropy function as the resolution refines. This type of convergence turns out to be the most important feature in the part of entropy theory leading to digitalization and data compression of topological dynamical systems.
The material in this section is based mainly on the paper [Downarowicz, 2005a].
The type of convergence
In this section we will try to understand what it means, for two monotone nets of real-valued functions converging to the same limit function, to converge “the same way,” or to represent the same “type of convergence.”
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- Entropy in Dynamical Systems , pp. 227 - 271Publisher: Cambridge University PressPrint publication year: 2011