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We show how to construct a topological Markov map of the interval whoseinvariant probability measure is the stationary law of a given stochastic chainof infinite order. In particular we caracterize the maps corresponding tostochastic chains with memory of variable length. The problem treated here isthe converse of the classical construction of the Gibbs formalism for Markovexpanding maps of the interval.

Chains of infinite order, chains with memory of variable length, and maps of the interval