Tame dynamics and robust transitivity chain-recurrence classes versus homoclinic classes
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- by C. Bonatti, S. Crovisier, N. Gourmelon and R. Potrie PDF
- Trans. Amer. Math. Soc. 366 (2014), 4849-4871 Request permission
Abstract:
One main task of smooth dynamical systems consists in finding a good decomposition into elementary pieces of the dynamics. This paper contributes to the study of chain-recurrence classes. It is known that $C^1$-generically, each chain-recurrence class containing a periodic orbit is equal to the homoclinic class of this orbit. Our result implies that in general this property is fragile.
We build a $C^1$-open set $\mathcal {U}$ of tame diffeomorphisms (their dynamics only splits into finitely many chain-recurrence classes) such that for any diffeomorphism in a $C^\infty$-dense subset of $\mathcal {U}$, one of the chain-recurrence classes is not transitive (and has an isolated point). Moreover, these dynamics are obtained among partially hyperbolic systems with one-dimensional center.
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Additional Information
- C. Bonatti
- Affiliation: CNRS - IMB, UMR 5584, Université de Bourgogne, 21004 Dijon, France
- Email: bonatti@u-bourgogne.fr
- S. Crovisier
- Affiliation: CNRS - LMO, UMR 8628. Université Paris-Sud 11, 91405 Orsay, France
- MR Author ID: 691227
- Email: sylvain.crovisier@math.u-psud.fr
- N. Gourmelon
- Affiliation: IMB, UMR 5251, Université Bordeaux 1, 33405 Talence, France
- Email: Nicolas.Gourmelon@math.u-bordeaux1.fr
- R. Potrie
- Affiliation: CMAT, Facultad de Ciencias, Universidad de la República, Uruguay
- MR Author ID: 863652
- ORCID: 0000-0002-4185-3005
- Email: rpotrie@cmat.edu.uy
- Received by editor(s): January 10, 2012
- Received by editor(s) in revised form: December 10, 2012
- Published electronically: May 8, 2014
- Additional Notes: The authors were partially supported by the ANR project DynNonHyp BLAN08-2 313375.
- © Copyright 2014 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 366 (2014), 4849-4871
- MSC (2010): Primary 37C20, 37D30, 37C29, 37G25
- DOI: https://doi.org/10.1090/S0002-9947-2014-06261-2
- MathSciNet review: 3217702