Finite-to-one zero-dimensional covers of dynamical systems (Q2004919)

This paper concerns a version of the Hurewicz theorem in the framework of dynamical systems: if \(f: X\to Y\) is a closed map between separable metric spaces and there exists \(k\geq 0\) such that \(\dim f^{-1}(y) \leq k\) for each \(y\in Y\), then \(\dim X \leq \dim Y + k\). The authors consider the following problem: what kind of dynamical systems \((X, f)\) can be covered by zero-dimensional dynamical systems via finite-to-one maps? For what kind of dynamical systems \((X, f)\), does there exist a dynamical system \((Z, \tilde{f})\) with a zero-dimensional compactum \(Z\) and a finite-to-one onto map \(p: Z\to X\) such that \(p \tilde{f} = f p\)? This paper gives a partial answer to this problem, generalizing \textit{J. Kulesza}'s result [Ergodic Theory Dyn. Syst. Journal Profile 15, No. 5, 939--950 (1995; Zbl 0882.54034)] (see an improved version in [\textit{Y. Ikegami} et al., Topology Appl. 160, No. 3, 564--574 (2013; Zbl 1295.54036)]). The main theorem states: let \(f: X\to X\) be a map on an \(n\)-dimensional compactum \(X\) such that \(f\) is zero-dimensional preserving (i.e., for any zero-dimensional closed \(D\subset X\), \(\dim f(D) \leq 0\)), the set \(\{x\in X| \dim f^{-1}(x) \leq 0\}\) is zero-dimensional and the set \(EP(f)\) of eventually periodic points is zero-dimensional. Then there exists a dense \(G_{\delta}\)-set \(H\) of \(X\) and a zero-dimensional cover \((Z,\tilde{f})\) of \((X,f)\) via an at most \(2^n\)-to-one onto map \(p\) such that \(EP(f) \subset H\) and \(|p^{-1}(x)| = 1\) for \(x\in H\). For the special case that \((X,f)\) is a positively expansive dynamical system with \(\dim X = n\), \((X,f)\) is covered by a subshift \((\Sigma, \sigma)\) of the shift map \(\sigma: \{1, 2, \ldots, k\}^\infty \to \{1, 2, \ldots, k\}^\infty \) via a \(2^n\)-to-one map. The authors also study some dynamical zero-dimensional decomposition theorems of spaces related to such maps.