Minimal Sets on Graphs and Dendrites

Abstract

The topological structure of minimal sets of continuous maps on graphs, dendrites and dendroids is studied. A full characterization of minimal sets on graphs and a partial characterization of minimal sets on dendrites are given. An example of a minimal set containing an interval on a dendroid is given.

Recent Developments in Dynamical Systems: Three Perspectives

“…In the interval case, these are finite and Cantor sets (see [Block & Coppel, 1992]), while for the circle case the circle itself can also be minimal. These results can be generalized to graphs, where minimal sets are characterized as finite sets, Cantor sets and also unions of finitely many pairwise disjoint circles [Balibrea et al, 2003, Mai, 2005.…”

“…In this case, besides partial results in [Balibrea et al, 2003], a complete characterization has been given recently in [Balibrea et al, 2009] as a consequence of a more general result based on the new notion of almost totally disconnected spaces. A space X is almost totally disconnected if the set of its degenerate components is dense in X.…”

Omega-limit sets in hereditarily locally connected continua

“…The main result, which generalizes Extension lemma from [28], is formulated in Proposition 11. From this we immediately obtain that the system ω X of all ω-limit sets of all continuous selfmaps of X coincides with the system W X for every hlc continuum X.…”

“…Section 3 deals with the existence of continuous surjections between special classes of compacta. We also give a generalization of the extension lemma from [28] to hlc continua. In Section 4 we state some sufficient conditions under which a countable union of nowhere dense ω-limit sets in a compact metric space is again an ω-limit set.…”

Iteration Theory: Dynamical Systems and Functional Equations

“…For some topological spaces of dimension one: I, S 1 , trees and graphs in [Balibrea et al, 2002a] a full characterization has been given. For other spaces of dimension one such that dendrites and dendroids, only partial characterizations are known (see the same paper).…”