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Entropy structures with continuous partitions of unity

Jérôme Carrand

Abstract

Using only continuous partitions of unity, we provide equivalent definitions for the metric, topological and topological tail entropies and pressures of a continuous self-map of a compact set, as well as their conditional versions. A tail variational principle for these new definitions is proved. We extend Downarowicz's notions of candidates and entropy structures to account for almost-increasing sequences of functions arising from the new definitions. Finally, we deduce a partial answer to a question raised by Newhouse.

Entropy structures with continuous partitions of unity

Abstract

Using only continuous partitions of unity, we provide equivalent definitions for the metric, topological and topological tail entropies and pressures of a continuous self-map of a compact set, as well as their conditional versions. A tail variational principle for these new definitions is proved. We extend Downarowicz's notions of candidates and entropy structures to account for almost-increasing sequences of functions arising from the new definitions. Finally, we deduce a partial answer to a question raised by Newhouse.

Paper Structure

This paper contains 18 sections, 45 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper and statement of results
  3. Topological and metric entropies
  4. Metric entropy via partitions of unity
  5. Topological entropy via continuous partitions of unity
  6. Equivalence of definitions
  7. Some other properties
  8. Topological and metric pressures
  9. Sequences with vanishing diameters
  10. Topological tail entropy and tail variational principle
  11. Conditional and tail topological entropies
  12. The case of vanishing diameters
  13. An alternative definition
  14. Topological tail pressure and variational principle
  15. Entropy structures and weak entropy structures
  16. ...and 3 more sections

Key Result

Theorem 1.1

For any topological dynamical system $(X,T)$ with finite topological entropy, it holds $\widetilde{h}_{\scriptsize{\hbox{\rm top}}}(T) = h_{\scriptsize{\hbox{\rm top}}}(T)$, and for any $T$-invariant measure $\mu$, $\widetilde{h}_{\mu}(T) = h_{\mu}(T)$. Furthermore, if $\Phi_n$ is a sequence of cont

Theorems & Definitions (104)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2: metric entropy
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 94 more