More on the Shift Dynamics–indecomposable Continua Connection

If X is a compact, locally connected metric space, f : X → X is a homeomorphism, and Q is a closed neighborhood of X, then Z = {p ∈ Q : f(p) ∈ Q for all integers n} is the permanent set for f on Q, and E = {p ∈ Q : there is some positive integer Np such that if n ≥ Np, then f−n(p) ∈ Q} is the entrainment set. In a previous paper, we began a study of the entrainment sets of topological horseshoes, and showed that, under mild conditions, the closure of the entrainment set for a topological horseshoe is “indecomposable–like” in that it admits a continuous map onto an indecomposable continuum. Furthermore, if f denotes the map associated with the topological horseshoe and K denotes the closure of the entrainment set for the horseshoe, then there is a map f̃ on the indecomposable continuum, denoted K̃, and a map h : K → K̃ such that h ◦ f̃ = f ◦ h, i.e., the dynamics of f on K factors over the dynamics of f̃ on K̃. Here we continue this study of the structure of entrainment sets of topological horseshoes and investigate the presence of invariant indecomposable continua contained in the closure of entrainment sets.