On homeomorphisms of ( [formula omitted], f) having topological entropy zero

Abstract

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.