A ug 2 01 4 PIECEWISE CONTRACTIONS DEFINED BY ITERATED FUNCTION SYSTEMS

Let φ1, . . . , φn : [0, 1] → (0, 1) be Lipschitz contractions. Let I = [0, 1), x0 = 0 and xn = 1. We prove that for Lebesgue almost every (x1, ..., xn−1) satisfying 0 < x1 < · · · < xn−1 < 1, the piecewise contraction f : I → I defined by x ∈ [xi−1, xi) 7→ φi(x) is asymptotically periodic. More precisely, f has at least one and at most n periodic orbits and the ω-limit set ωf (x) is a periodic orbit of f for every x ∈ I.