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On mixing and dense periodicity on spaces with a free arc

Dominik Kwietniak, Filip Wierzbowski

Abstract

We study the dynamics of continuous maps on compact metric spaces containing a free interval (an open subset homeomorphic to the interval $(0,1)$). We provide a new proof of a result of M. Dirbák, Ľ. Snoha, V. Špitalský [Ergodic Theory Dynam. Systems, vol. 33 (2013), no. 6, pp. 1786--1812] saying that every continuous and transitive, but non-minimal map of a space with a free interval is relatively mixing, non-invertible, has positive topological entropy, and dense periodic points. The key simplification comes from short proofs of two facts. The first says that every weakly mixing map of a space with a free interval must be mixing and have positive entropy. The second says that a transitive but not minimal map of a space with a free interval has dense periodic points and is non-invertible.

On mixing and dense periodicity on spaces with a free arc

Abstract

We study the dynamics of continuous maps on compact metric spaces containing a free interval (an open subset homeomorphic to the interval ). We provide a new proof of a result of M. Dirbák, Ľ. Snoha, V. Špitalský [Ergodic Theory Dynam. Systems, vol. 33 (2013), no. 6, pp. 1786--1812] saying that every continuous and transitive, but non-minimal map of a space with a free interval is relatively mixing, non-invertible, has positive topological entropy, and dense periodic points. The key simplification comes from short proofs of two facts. The first says that every weakly mixing map of a space with a free interval must be mixing and have positive entropy. The second says that a transitive but not minimal map of a space with a free interval has dense periodic points and is non-invertible.
Paper Structure (4 sections, 11 theorems, 19 equations, 1 figure)

This paper contains 4 sections, 11 theorems, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Terminology and notation
  3. When does weak mixing imply mixing?
  4. Transitivity and dense periodicity

Key Result

Lemma 1

If $g\colon [0,1] \to [0,1]$ is a continuous surjection, then for any choice of $0 \leqslant c < d \leqslant 1$ there exist $0 \leqslant a < b \leqslant 1$ such that $g([a,b]) = [c,d]$.

Figures (1)

  • Figure 1: Example illustration of the graphs $\Gamma_n$ and $\Gamma_m$, and points $s$, $t$, $u$ used in the argument.

Theorems & Definitions (19)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3: Horseshoe lemma
  • Lemma 4
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • ...and 9 more