Asymptotic behaviour of iterated piecewise monotone maps

Abstract

Abstract In this paper the asymptotic behaviour of piecewise monotone functions f : I → I with a finite number of discontinuities is studied (where I ⊆ ℝ is a compact interval). It is shown that there is a finite number of f -almost-invariant subsets C 1 ,…, C r , R 1 ,…, R s , where each C i is a disjoint union of closed intervals and each R j is a Cantor-like subset of I , such that if x is a ‘typical’ point in I (in a topological sense) then exactly one of the following three possibilities will happen: (1) { f n ( x )} n ≥ 0 eventually ends up in some C i . (2) { f n ( x )} n ≥ 0 is attracted to some R j . (3) { f n ( x ): n ≥ 0} is contained in an open, invariant set Z ⊆ I , which is such that for each n ≥ 1 f n is monotone and continuous on each connected component of Z . Moreover, f acts topologically transitively on each C i and minimally on each R j . Furthermore, it is shown how the sets C 1 ,…, C r , R 1 ,…, R s can be constructed. Finally, our results are applied to some examples.