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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: [email protected], [email protected])
CHRIS GOOD
Affiliation:
School of Mathematics and Statistics, University of Birmingham, Birmingham B15 2TT, UK (email: [email protected], [email protected])
ROBIN KNIGHT
Affiliation:
Mathematical Institute, University of Oxford, Oxford OX1 3LB, UK (email: [email protected])
BRIAN E. RAINES
Affiliation:
Department of Mathematics, Baylor University, Waco, TX 76798–7328, USA (email: [email protected])
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Abstract

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.

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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References

[1]Balibrea, F. and La Paz, C.. A characterization of the ω-limit sets of interval maps. Acta Math. Hungar. 88(4) (2000), 291–300.CrossRefGoogle Scholar
[2]Brucks, K. and Bruin, H.. Subcontinua of inverse limit spaces of unimodal maps. Fund. Math. 160(3) (1999), 219–246.CrossRefGoogle 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), 107–131.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