Symmetric Itinerary Sets

We consider a one parameter family of dynamical systems W :[0, 1] -> [0, 1] constructed from a pair of monotone increasing diffeomorphisms Wsub(i), such that Wsub(i)(inverse): [0, 1] -> [0, 1], (i = 0, 1). We characterise the set of symbolic itineraries of W using an attractor of an iterated closed relation,in the terminology of McGehee, and prove that there is a member of the family for which is symmetrical.