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 < delta < epsilon < 1 and let phi(i) : {[}0, 1] -> (0, 1), 1 <= i <= n, be a family of C-2 maps whose ranges phi(1) ({[}0, 1]), ..., phi(n)({[}0, 1]) are pairwise disjoint and delta < vertical bar D phi(i)(x)vertical bar < epsilon for every x is an element of (0, 1). Let 0 < x(1) < ... < x(n-1) < 1 and let I-1, ..., I-n be a partition of the interval {[}0, 1) into subintervals Ii having interior (x(i-1), x(i)), where x(0) = 0 and x(n) = 1. Let f(x1), ..., x(n-1) be the map given by x bar right arrow phi(i)(x) if x is an element of I-i, for 1 <= i <= n. Among other results we prove that for Lebesgue almost every point (x(1), ..., x(n-1)), the piecewise contraction f(x1, ..., xn-1) is asymptotically periodic. (AU)