Eventual colorings of homeomorphisms

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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 → X with zero-dimensional set of periodic points. In particular, we show that if dim X < ∞, then there is a natural number p, which depends on dim X, and X can be divided into two closed regions C1 and C2 such that for each point x ∈ X, the orbit {fk(x)}k=0∞ of x goes back and forth between C1 − C2 and C2 − C1 within the time p.

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