Endpoints of Inverse Limit Spaces and Dynamics

Inhomogeneities in chainable continua

“…It gives a characterization of interval maps that are non-retractable along ω(C), through the dynamical properties of the maps. [8] give a characterization of endpoints in lim ← − {I, f } in terms of the dynamics of f . They prove that if f has finitely many critical points, a dense orbit, and if ω(C) ∩ C = ∅, then there are no endpoints.…”

“…is in lim ← − {I, f } and they are endpoints (see [8,Theorem 2.9] for the detailed argument). We choose an arbitrarily small interval L 0 such that…”

“…. ), where x ∈ I is a fixed point of f , is an endpoint (see [8,Example 3], for the detailed argument), which gives countably many endpoints. Furthermore, all points (x 0 , x 1 , x 2 , .…”

“…Our study is motivated by results on the structure of local inhomogeneities of unimodal inverse limit spaces (see e.g. [8,38,24,2]), and by the lack of basic tools to study inhomogeneities of indecomposable onedimensional continua beyond the unimodal setting.…”

“…If f : I → I is unimodal, then lim ← − {I, f } contains an endpoint if and only if c is recurrent, i.e., c ∈ ω(c). Moreover, if the orbit of c is infinite, then lim ← − {I, f } has uncountably many endpoints [8], [24].…”