On homeomorphisms of (I, f) having topological entropy zero

Let f : I → I be a continuous function where I is the unit interval. Let (I, f) be the inverse limit space obtained from the inverse sequence all of whose maps are f and all of whose spaces are I. This paper addresses the question of when (I, f) has the property that every homeomorphism of (I, f) has zero topological entropy. An obvious necessary condition for this is that f itself has zero topological entropy. In this paper it is proved that if f is piecewise monotone and has only finitely many periods, then every homeomorphism of (I, f) has zero entropy.