Asymptotically periodic piecewise contractions of the interval

We consider the iterates of a generic injective piecewise contraction of the interval defined by a finite family of contractions. Let 0 < δ <  < 1 and let φ_i : [0, 1] → (0, 1), 1 ⩽ i ⩽ n, be a family of C² maps whose ranges φ₁([0, 1]), ···, φ_n([0, 1]) are pairwise disjoint and δ < |Dφ_i(x)| <  for every x ∈ (0, 1). Let 0 < x₁<···<x_n−1 < 1 and let I₁, ..., I_n be a partition of the interval [0, 1) into subintervals I_i having interior (x_i−1, x_i), where x₀ = 0 and x_n = 1. Let $f_{x_1,\ldots,x_{n-1}}$ be the map given by x ↦ φ_i(x) if x ∈ I_i, for 1 ⩽ i ⩽ n. Among other results we prove that for Lebesgue almost every point (x₁, ..., x_n−1), the piecewise contraction $f_{x_1,\ldots,x_{n-1}}$ is asymptotically periodic.