Eventual colorings of homeomorphisms

Abstract

In this paper, we study some dynamical properties of fixed-point free homeomorphisms of separable metric spaces. For each natural number $p$, we define eventual colorings within $p$ of homeomorphisms which are generalized notions of colorings of fixed-point free homeomorphisms, and we investigate the eventual coloring number $C(f,p)$ of a fixed-point free homeomorphism $f: X \to X$ with zero-dimensional set of periodic points. In particular, we show that if $\dim X$ < $\infty$, then there is a natural number $p$, which depends on $\dim X$, and $X$ can be divided into two closed regions $C_{1}$ and $C_{2}$ such that for each point $x\in X$, the orbit $\{f^{k}(x)\}_{k=0}^{\infty}$ of $x$ goes back and forth between $C_1-C_2$ and $C_2-C_1$ within the time $p$.