Asymptotic behaviour of iterated piecewise monotone maps

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 C₁,…, C_r, R₁,…, 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) {fⁿ (x)}_{n ≥ 0} eventually ends up in some C_i.(2) {fⁿ (x)}_{n ≥ 0} is attracted to some R_j.(3) {fⁿ (x): n ≥ 0} is contained in an open, invariant set Z ⊆ I, which is such that for each n ≥ 1 fⁿ 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 C₁,…, C_r, R₁,…, R_s can be constructed. Finally, our results are applied to some examples.