Every nondegenerate Peano continuum admits a pure mixing selfmap

TL;DR

This work proves that every nondegenerate Peano continuum supports a self-map that is mixing but not exact, with a dense set of periodic points (pure mixing). The authors achieve this via a two-step construction: a map $f:X\to[0,1]$ ensuring open-sets map onto sets with interior, followed by a detailed inductive construction of $g:[0,1]\to X$ so that the composition $h=g\circ f$ is mixing and has dense periodic points, yet remains non-invertible. The paper also analyzes entropy aspects, showing the constructed pure mixing maps typically exhibit large or infinite topological entropy, and discusses open questions about entropy infima and the existence of mixing homeomorphisms on Peano continua. Overall, the results advance understanding of chaotic dynamics on Peano continua by separating mixing from exactness and highlighting entropy as a key distinguishing feature.

Abstract

We prove that every Peano continuum (a space that is a continuous image of $[0,1]$) admits a topologically mixing but not exact map. The constructed map has a dense set of periodic points.

Theorems & Definitions (27)

• Theorem 1.1
• Definition 2.1
• Theorem 2.2
• proof
• Theorem 2.3
• Remark 2.4
• Theorem 2.5: Tietze Extension Theorem
• Lemma 3.1
• proof
• Lemma 3.2