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.
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