Ergodic Properties and Invariant Measures

Abstract

In this chapter we want to study the typical behaviour of orbits. Up till now we have seen that the topological behaviour of interval maps is quite well understood: for example if f is a unimodal interval map with negative Schwarzian derivative and such that the fixed point on the boundary is repelling, then by Guckenheimer’s theorem, see Theorem III.4.1, there are three possibilities: 1 f has a periodic attractor and then the basin of this attractor is a dense set in the interval; 2 f is infinitely renormalizable and then there exists a corresponding solenoidal Cantor set on which f acts as an adding machine, and, furthermore, a dense set of points is attracted by this Cantor set; 3 f is finitely often renormalizable and f is transitive on some finite union of intervals Λ: there exists a dense orbit in Λ. A dense set of points is attracted to A and periodic points appear densely in Λ.