Periodic points, compactifications and eventual colorings of maps (Q1944340)

All spaces are assumed to be separable metric. Suppose that \(X\) is a space and \(f:X\to X\) is a fixed-point free map. Given a number \(p\in \mathbb N\), a set \(C\subset X\) is said to be \textit{eventually colored within \(p\)} if \(\bigcap_{i=0}^p f^{-i}(C)=\emptyset\). The map \(f\) is called \textit{eventually 2-colorable} if there exists a number \(p\in \mathbb N\) and open (or closed) sets \(A,B\subset X\) such that \(X=A\cup B\) while both \(A\) and \(B\) are eventually colored within \(p\). A point \(x\in X\) is called \textit{periodic} if \(f^p(x)=x\) for some \(p\in \mathbb N\). The following facts are the main results of the paper. \noindent { 1. Theorem.} Assume that \(X\) is a finite-dimensional separable metric space and \(f:X\to X\) is a fixed-point free map with a zero-dimensional set of periodic points. If there exists \(n\in \mathbb N\) such that \(|f^{-1}(x)|\leq n\) for all \(x\in X\), then \(f\) is eventually 2-colorable. \noindent { 2. Theorem.} Assume that \(X\) is a locally compact finite-dimensional separable metric space and \(f:X\to X\) is a fixed-point free map with a zero-dimensional set of periodic points. Then \(f\) is eventually 2-colorable.