Homtervals

Abstract

So far, we have analyzed in some detail the relationship between itineraries and points in [-1,1]. We now ask the question of how many points can have the same itinerary. The answer to this question will be relevant for the sections on topological conjugacy, on sensitive behavior and on ergodic properties. It turns out that in the case of S-unimodal maps there are essentially two possibilities for the relationship between itineraries and points. If f is S-unimodal and has no stable periodic orbit, then we shall see that two different points always have different itineraries. On the other hand, if f is S-unimodal and has a stable periodic orbit, then almost all points (in the Lebesgue sense) have itineraries which are eventually periodic with the periodic part equal to the itinerary of a point of the periodic orbit (except in the superstable case).