Minimal spaces with cyclic group of homeomorphisms

TL;DR

This paper investigates the entropy of functional envelopes in topological dynamical systems, introduces Slovak spaces with cyclic homeomorphism groups, and constructs examples with surprising entropy properties, including non-uniqueness of the circle.

Contribution

It introduces Slovak spaces with cyclic homeomorphism groups, providing new examples of dynamical systems with unique entropy characteristics and resolving longstanding open problems.

Findings

Entropies are infinite for zero-dimensional spaces unless the system is equicontinuous.

Existence of systems with positive entropy but zero entropy in their functional envelopes.

The circle is not unique as a non-degenerate continuum with only invertible minimal transformations.

Abstract

There are two main subjects in this paper. 1) For a topological dynamical system $(X,T)$ we study the topological entropy of its "functional envelopes" (the action of $T$ by left composition on the space of all continuous self-maps or on the space of all self-homeomorphisms of $X$). In particular we prove that for zero-dimensional spaces $X$ both entropies are infinite except when $T$ is equicontinuous (then both equal zero). 2) We call $Slovak$ $space$ any compact metric space whose homeomorphism group is cyclic and generated by a minimal homeomorphism. Using Slovak spaces we provide examples of (minimal) systems $(X,T)$ with positive entropy, yet, whose functional envelope on homeomorphisms has entropy zero (answering a question posed by Kolyada and Semikina). Finally, also using Slovak spaces, we resolve a long standing open problem whether the circle is a unique non-degenerate continuum admitting minimal continuous transformations but only invertible: No, some Slovak spaces are such, as well.