M ay 2 01 6 TOPOLOGICAL DYNAMICS OF PIECEWISE λ-AFFINE MAPS

Let −1 < λ < 1 and f : [0, 1) → R be a piecewise λ-affine map, that is, there exist points 0 = c0 < c1 < · · · < cn−1 < cn = 1 and real numbers b1, . . . , bn such that f(x) = λx + bi for every x ∈ [ci−1, ci). We prove that, for Lebesgue almost every δ ∈ R, the map fδ = f + δ (mod 1) is asymptotically periodic. More precisely, fδ has at most 2n periodic orbits and the ω-limit set of every x ∈ [0, 1) is a periodic orbit.