The shift dynamics–indecomposable continua connection

For F a homeomorphism on a compact, locally connected, separable metric space having an invariant isolated set A such that F|A factors over the shift σ on M-symbols, we prove, with the addition of mild conditions, that A is contained in an invariant continuum K which (i) is the closure of the entrainment set of A, and (ii) is such that F|L factors over a homeomorphism on an indecomposable continuum K. Continua admitting homeomorphisms which factor over homeomorphisms on indecomposable continua are “indecomposable-like” in many respects. In particular, such continua do not admit continuos maps onto locally connected continua. Roughly speaking, indecomposability is the vehicle a dynamical system on a locally connected space uses to make the transition from an invariant subset having shift dynamics (thus the subset is quite disconnected) to the “rest” of the space.