Indecomposability and the structure of periodic orbits for interval maps

Suppose f :[a,b]→[a,b] is continuous. Barge and Martin, and Ingram have shown that if the inverse limit of {[a,b],f} is hereditarily decomposable, then the period of every periodic orbit of f is a power of two. We will elaborate on the structure of these orbits, and, assuming f is a Markov map whose partition is a single periodic orbit, give necessary and sufficient conditions for the inverse limit to be (1) decomposable and (2) hereditarily decomposable.