Statistical convergent topological sequence entropy maps of the circle

Abstract

A continuous map f of the interval is chaotic iff there is an increasing of nonnegative integers T such that the topological sequence entropy off relative to T, hT(f), is positive. On the other hand, for any increasing sequence of nonnegative integers T there is a chaotic map f of the interval such that hT(f)=0. We prove that the same results hold for maps of the circle. We also prove some preliminary results concerning statistical convergent topological sequence entropy for maps of general compact metric spaces.