A mixing-like property and inexistence of invariant foliations for minimal diffeomorphisms of the 2-torus
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- by Alejandro Kocsard and Andrés Koropecki PDF
- Proc. Amer. Math. Soc. 137 (2009), 3379-3386 Request permission
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.References
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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
- MR Author ID: 856885
- Email: koro@mat.uff.br
- 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
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2515407