Topology and its Applications 155 (2008) 1237–1255 www.elsevier.com/locate/topol Omega-limit sets in hereditarily locally connected continua Vladimír Špitalský 1 Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01 Banská Bystrica, Slovakia Received 1 August 2007; received in revised form 5 March 2008; accepted 5 March 2008 Abstract In this paper we give a topological characterization of ω-limit sets in hereditarily locally connected continua. Moreover, we characterize also orbit-enclosing ω-limit sets in these spaces.  2008 Elsevier B.V. All rights reserved. MSC: primary 37B05, 37B20, 37B40; secondary 54H20 Keywords: Omega-limit set; Continuum; Hereditarily locally connected continuum 1. Introduction By a dynamical system we mean a pair (X, f ) where X is a compactum (i.e., a compact metric space) and f is a continuous selfmap of X. The f -trajectory of a point x ∈ X is the sequence (f n (x)) ∞ n=0 , where f 0 is the identity on X and f n+1 = f ◦ f n for any n  0. The ω-limit set of f at the point x , denoted by ω f (x), is the set of all limit points of the trajectory of x . The system of all ω-limit sets of the system (X, f ) is denoted by ω f . Finally, by ω X we denote the union of ω f over all continuous selfmaps f of X. The topological characterization of the system ω X in the interval was given in [1] (see also [2]). It is proved there that a nonempty subset M of the interval is an ω-limit set of some continuous selfmap of the interval if and only if it is either a finite union of nondegenerate closed subintervals or a nowhere dense closed set. (Let us remark that using this result one can obtain the folklore fact that a nonempty subset M of a zero-dimensional compactum X is in ω X if and only if it is either a finite set or a Cantor set or a nowhere dense closed set.) In [3] a topological characterization of ω-limit sets in the circle was given. This was recently generalized in [4] where ω X is characterized for every graph X. (A graph is a continuum which can be written as the union of finitely many arcs any two of which are either disjoint or intersect only in one or both of their end points. A continuum is a nonempty connected compactum.) The characterization is as follows: Theorem. (See [4].) Let X be a graph. Then a nonempty subset M of X is an ω-limit set of some continuous selfmap of X if and only if M is E-mail address: spitalsk@fpv.umb.sk. 1 The author was supported by Science and Technology Assistance Agency under the contract No. APVT-20-016304. 0166-8641/$ – see front matter  2008 Elsevier B.V. All rights reserved. doi:10.1016/j.topol.2008.03.005 brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by Elsevier - Publisher Connector