Commutativity and non-commutativity of the topological sequence entropy

Abstract

In this paper we study the commutativity property for topological sequence entropy. We prove that if $X$ is a compact metric space and $f,g: X\rightarrow X$ are continuous maps then $h _A(f\circ g)=h_A(g\circ f)$ for every increasing sequence $A$ if $X=[0,1]$, and construct a counterexample for the general case. In the interim, we also show that the equality $h_A(f)=h_A(f\vert _{\cap _{n\ge 0}f^n(X)})$ is true if $X=[0,1]$ but does not necessarily hold if $X$ is an arbitrary compact metric space.