Invariant measures for piecewise continuous maps

Abstract

We say that f:[0,1]→[0,1] is a piecewise continuous interval map if there exists a partition 0=x0<x1<⋯<xd<xd+1=1 of [0,1] such that f|(xi−1,xi) is continuous and the lateral limits w0+=limx→0+⁡f(x), wd+1−=limx→1−⁡f(x), wi−=limx→xi−⁡f(x) and wi+=limx→xi+⁡f(x) exist for each i. We prove that every piecewise continuous interval map without connections admits an invariant Borel probability measure. We also prove that every injective piecewise continuous interval map with no connections and no periodic orbits is topologically semiconjugate to an interval exchange transformation.