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Rokhlin towers and closing for flows on T2

Published online by Cambridge University Press:  19 September 2008

C. R. Carroll
C. R. Carroll
Affiliation:
Department of Mathematics, Northwestern University, Evanston, IL 60208, USA
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Abstract

This paper considers the closing problem, r≥2, for flows on T2 with finitely many singularities. We study the action of such flows on circles transverse to the flow, using Katznelson and Ornstein's picture of rigid rotations as Rokhlin towers. We are able to obtain closing on purely topological grounds for a large class of flows by means of perturbations which simply twist a flow along an embedded transverse circle (twist-perturbations). However, we give an example of a flow (with finitely many singularities) for which no small twist-perturbation yields closing; indeed such perturbations either leave the non-wandering set unchanged or else collapse it to the set of singularities.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 12 , Issue 4 , December 1992 , pp. 683 - 706
Copyright
Copyright © Cambridge University Press 1992

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References

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