Commutativity and non-commutativity of topological sequence entropy

In this paper we study the commutativity property for topological sequence entropy. We prove that ii X is a compact metric space and f, g : X --> X ar e continuous maps then h(A)(f o g) = hA(g o f) for every increasing sequenc e 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 \(n)(boolean A ND n greater than or equal to 0f)(X)) is true if X = [0, 1] but does not ne cessarily hold if X is an arbitrary compact metric space.