Relativization of complexity and sensitivity

Abstract

First notions of relative complexity function and relative sensitivity are introduced. It turns out that for any open factor map $\pi: (X, T)\rightarrow (Y, S)$ between topological dynamical systems with minimal $(Y, S),\ \pi$ is positively equicontinuous if and only if the relative complexity function is bounded for each open cover of $X$; and that any non-trivial weakly mixing extension is relatively sensitive. Moreover, a relative version of the notable result that any $M$-system is sensitive if it is not minimal is obtained. Then notions of relative scattering and relative Mycielski's chaos are introduced. A relative disjointness theorem involving relative scattering is given. A relative version of the well-known result that any non-trivial scattering topological dynamical system is Li–Yorke chaotic is proved.