Hostname: page-component-7dc689bd49-6fmns Total loading time: 0 Render date: 2023-03-20T09:39:11.886Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

A characterization of ω-limit sets in shift spaces

Published online by Cambridge University Press:  26 February 2009

School of Mathematics and Statistics, University of Birmingham, Birmingham B15 2TT, UK (email:,
School of Mathematics and Statistics, University of Birmingham, Birmingham B15 2TT, UK (email:,
Mathematical Institute, University of Oxford, Oxford OX1 3LB, UK (email:
Department of Mathematics, Baylor University, Waco, TX 76798–7328, USA (email:


A set Λ is internally chain transitive if for any x,y∈Λ and ϵ>0 there is an ϵ-pseudo-orbit in Λ between x and y. In this paper we characterize all ω-limit sets in shifts of finite type by showing that, if Λ is a closed, strongly shift-invariant subset of a shift of finite type, X, then there is a point zX with ω(z)=Λ if and only if Λ is internally chain transitive. It follows immediately that any closed, strongly shift-invariant, internally chain transitive subset of a shift space over some alphabet ℬ is the ω-limit set of some point in the full shift space over ℬ. We use similar techniques to prove that, for a tent map f, a closed, strongly f-invariant, internally chain transitive subset of the interval is the ω-limit set of a point provided it does not contain the image of the critical point. We give an example of a sofic shift space Z𝒢 (a factor of a shift space of finite type) that is not of finite type that has an internally chain transitive subset that is not the ω-limit set of any point in Z𝒢.

Research Article
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


[1]Balibrea, F. and La Paz, C.. A characterization of the ω-limit sets of interval maps. Acta Math. Hungar. 88(4) (2000), 291300.CrossRefGoogle Scholar
[2]Brucks, K. and Bruin, H.. Subcontinua of inverse limit spaces of unimodal maps. Fund. Math. 160(3) (1999), 219246.Google Scholar
[3]Collet, P. and Eckmann, J.-P.. Iterated Maps on the Interval as Dynamical Systems. Birkhäuser, Boston, MA, 1980.Google Scholar
[4]Guckenheimer, J. and Holmes, P.. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Applied Mathematical Sciences, 42). Springer, New York, 1990. (Revised and corrected reprint of the 1983 original.)Google Scholar
[5]Smith, H. L., Zhao, X.-Q. and Hirsch, M. W.. Chain transitivity, attractivity and strong repelors for semidynamical systems. J. Dynam. Differential Equations 13(1) (2001), 107131.Google Scholar
[6]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54). Cambridge University Press, Cambridge, 1995. (With a supplementary chapter by Katok and Leonardo Mendoza.)CrossRefGoogle Scholar
[7]Lind, D. and Marcus, B.. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar