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We prove that for Lebesgue almost every $(x_1,...,x_{n-1})$ satisfying $0<x_1<\\cdots <x_{n-1}<1$, the piecewise contraction $f:I\\to I$ defined by $x\\in [x_{i-1},x_i)\\mapsto \\phi_i(x)$ is asymptotically periodic. 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