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We present a family of continuous piecewise linear maps of the unit interval into itself that are all chaotic in the sense of Li and Yorke [‘Period three implies chaos’, Amer. Math. Monthly82 (1975), 985–992] and for which almost every point (in the sense of Lebesgue) in the unit interval is an eventually periodic point of period 
$p,p\geq 3$, for a member of the family.
