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A mixing-like property and inexistence of invariant foliations for minimal diffeomorphisms of the 2-torus


Authors: Alejandro Kocsard and Andrés Koropecki
Journal: Proc. Amer. Math. Soc. 137 (2009), 3379-3386
MSC (2000): Primary 37E30, 37B05
DOI: https://doi.org/10.1090/S0002-9939-09-09903-1
Published electronically: May 6, 2009
MathSciNet review: 2515407
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Abstract: We consider diffeomorphisms in $ \overline{\mathcal{O}}^\infty(\mathbb{T}^2)$, the $ C^\infty$-closure of the conjugancy class of translations of $ \mathbb{T}^2$. By a theorem of Fathi and Herman, a generic diffeomorphism in that space is minimal and uniquely ergodic. We define a new mixing-type property, which takes into account the ``directions'' of mixing, and we prove that generic elements of $ \overline{\mathcal{O}}^\infty(\mathbb{T}^2)$ satisfy this property. As a consequence, we obtain a residual set of strictly ergodic diffeomorphisms without invariant foliations of any kind. We also obtain an analytic version of these results.


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Additional Information

Alejandro Kocsard
Affiliation: Instituto de Matemática, Universidade Federal Fluminense, Rua Mário Santos Braga S/N, 24020-140 Niteroi, RJ, Brazil
Email: alejo@impa.br

Andrés Koropecki
Affiliation: Instituto de Matemática, Universidade Federal Fluminense, Rua Mário Santos Braga S/N, 24020-140 Niteroi, RJ, Brazil
Email: koro@mat.uff.br

DOI: https://doi.org/10.1090/S0002-9939-09-09903-1
Received by editor(s): March 26, 2008
Received by editor(s) in revised form: January 25, 2009
Published electronically: May 6, 2009
Additional Notes: The authors were supported by CNPq-Brazil.
Communicated by: Jane M. Hawkins
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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