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A characterization of ω-limit sets in shift spaces

Published online by Cambridge University Press:  26 February 2009

ANDREW BARWELL,
CHRIS GOOD,
ROBIN KNIGHT and
BRIAN E. RAINES
ANDREW BARWELL
Affiliation:
School of Mathematics and Statistics, University of Birmingham, Birmingham B15 2TT, UK (email: barwella@for.mat.bham.ac.uk, c.good@bham.ac.uk)
CHRIS GOOD
Affiliation:
School of Mathematics and Statistics, University of Birmingham, Birmingham B15 2TT, UK (email: barwella@for.mat.bham.ac.uk, c.good@bham.ac.uk)
ROBIN KNIGHT
Affiliation:
Mathematical Institute, University of Oxford, Oxford OX1 3LB, UK (email: knight@maths.ox.ac.uk)
BRIAN E. RAINES
Affiliation:
Department of Mathematics, Baylor University, Waco, TX 76798–7328, USA (email: brian_raines@baylor.edu)
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Abstract

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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 z∈X 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.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 1 , February 2010, pp. 21 - 31
Copyright
Copyright © Cambridge University Press 2009

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