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Weak mixing disc and annulus diffeomorphisms with arbitrary Liouville rotation number on the boundary
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2005 (English)In: Annales Scientifiques de l'Ecole Normale Supérieure, ISSN 0012-9593, Vol. 38, no 3, 339-364 p.Article in journal (Refereed) Published
Abstract [en]

Let M be an m-dimensional differentiable manifold with a nontrivial circle action S = {S-t}(t is an element of R), St+1 = S-t, preserving a smooth volume mu. For any Liouville number alpha we construct a sequence of area-preserving diffeomorphisms H-n such that the sequence H-n circle S-alpha circle H-n(-1) converges to a smooth weak mixing diffeomorphism of M. The method is a quantitative version of the approximation by conjugations construction introduced in [Trans. Moscow Math. Soc. 23 (1970) 1]. For m = 2 and M equal to the unit disc D-2 = {x(2) + y(2) <= 1} or the closed annulus A = T x [0, 1] this result proves the following dichotomy: alpha is an element of R \ Q is Diophantine if and only if there is no ergodic diffeomorphism of M whose rotation number on the boundary equals alpha (on at least one of the boundaries in the case of A). One part of the dichotomy follows from our constructions, the other is an unpublished result of Michael Herman asserting that if alpha is Diophantine, then any area preserving diffeomorphism with rotation number alpha on the boundary (on at least one of the boundaries in the case of A) displays smooth invariant curves arbitrarily close to the boundary which clearly precludes ergodicity or even topological transitivity.

Place, publisher, year, edition, pages
2005. Vol. 38, no 3, 339-364 p.
Keyword [en]
flows
Identifiers
URN: urn:nbn:se:kth:diva-15067ISI: 000232187800001OAI: oai:DiVA.org:kth-15067DiVA: diva2:333108
Note
QC 20100525Available from: 2010-08-05 Created: 2010-08-05Bibliographically approved

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