Graph Theoretic Structure of Maps of the Cantor Space

TL;DR

This paper develops graph theoretic techniques to analyze the structure and dynamics of maps on the Cantor space, providing new characterizations of conjugacy classes and revealing properties of typical elements.

Contribution

It introduces unifying graph theoretic methods to characterize conjugacy and typical behavior in the space of homeomorphisms and self-maps of the Cantor space.

Findings

Characterization of conjugacy classes of homeomorphisms.

Existence of a comeager conjugacy class with specific properties.

Most self-maps of the Cantor space are conjugate to each other within a comeager set.

Abstract

In this paper we develop unifying graph theoretic techniques to study the dynamics and the structure of the space of homeomorphisms and the space of self-maps of the Cantor space. Using our methods, we give characterizations which determine when two homeomorphisms of the Cantor space are conjugate to each other. We also give a new characterization of the comeager conjugacy class of the space of homeomorphisms of the Cantor space. The existence of this class was established by Kechris and Rosendal and a specific element of this class was described concretely by Akin, Glasner and Weiss. Our characterization readily implies many old and new dynamical properties of elements of this class. For example, we show that no element of this class has a Li-Yorke pair, implying the well known Glasner-Weiss result that there is a comeager subset of homeomorphism space of the Cantor space each element of which has topological entropy zero. Our analogous investigation in the space of continuous self-maps of the Cantor space yields a surprising result: there is a comeager subset of the space of self-maps of the Cantor space such that any two elements of this set are conjugate to each other by an homeomorphism. Our description of this class also yields many old and new results concerning dynamics of a comeager subset of the space of continuous self-maps of the Cantor space.