Symbolic dynamics of piecewise contractions

A map is a piecewise contraction of n intervals (n-PC) if there exist and a partition of into intervals such that for every (). An infinite word over the alphabet is a natural coding of f if there exists such that whenever . We prove that if is a natural coding of an injective n-PC, then some infinite subword of is either periodic or isomorphic to a natural coding of a topologically transitive m-interval exchange transformation (m-IET), where . Conversely, every natural coding of a topologically transitive n-IET is also a natural coding of some injective n-PC.