Topological Speedups

TL;DR

This paper explores when one minimal homeomorphism on a Cantor space can be considered a speedup of another, providing a topological analogue to a classical measure-theoretic speedup theorem.

Contribution

It establishes conditions under which one minimal homeomorphism is a speedup of another, extending the concept to the topological setting.

Findings

Provides a topological analogue of the measure-theoretic speedup theorem.

Establishes conditions for when one minimal homeomorphism is a speedup of another.

Serves as a topological version of orbit equivalence results.

Abstract

Given a dynamical system $(X,T)$ one can define a speedup of $(X,T)$ as another dynamical system conjugate to $S:X\rightarrow X$ where $S(x)=T^{p(x)}(x)$ for some function $p:X\rightarrow\mathbb{Z}^{+}$. In $1985$ Arnoux, Ornstein, and Weiss showed that any aperiodic, not necessarily ergodic, measure preserving system is isomorphic to a speedup of any ergodic measure preserving system. In this paper we study speedups in the topological category. Specifically, we consider minimal homeomorphisms on Cantor spaces. Our main theorem gives conditions on when one such system is a speedup of another. Furthermore, the main theorem serves as a topological analogue of the Arnoux, Ornstein, and Weiss speedup theorem, as well as a 'one-sided" orbit equivalence theorem.