Dynamical decomposition theorems of homeomorphisms with zero-dimensional sets of periodic points

In this note, we study some dynamical decomposition theorems of spaces related to given homeomorphisms. First, we prove that if f:X→Xf:X→X is a homeomorphism of an n-dimensional separable metric space X with zero-dimensional set of periodic points, then X can be decomposed into a zero-dimensional bright space of f except n   times and an (n−1)(n−1)-dimensional dark space of f except n times, and also by use of dark spaces, we prove some decomposition theorems of X related to dimension theory and dynamical systems. Finally, we study dynamical decompositions of continuum-wise expansive homeomorphisms.