Finite-to-one zero-dimensional covers of dynamical systems

Abstract

In this paper, we study the existence of finite-to-one zero-dimensional covers of dynamical systems. Kulesza showed that any homeomorphism 𝑓 : 𝑋 → 𝑋 on an 𝑛-dimensional compactum 𝑋 with zero-dimensional set 𝑃(𝑓) of periodic points can be covered by a homeomorphism on a zero-dimensional compactum via an at most (𝑛 + 1)^𝑛-to-one map. Moreover, Ikegami, Kato and Ueda showed that in the theorem of Kulesza, the condition of at most (𝑛 + 1)^𝑛-to-one map can be strengthened to the condition of at most 2^𝑛-to-one map. In this paper, we will show that the theorem is also true for more general maps except for homeomorphisms. In fact we prove that the theorem is true for a class of maps containing two-sided zero-dimensional maps. For the special case, we give a theorem of symbolic extensions of positively expansive maps. Finally, we study some dynamical zero-dimensional decomposition theorems of spaces related to such maps.