Tame dynamics and robust transitivity chain-recurrence classes versus homoclinic classes

Bonatti, C.; Crovisier, S.; Gourmelon, N.; Potrie, R.

doi:10.1090/S0002-9947-2014-06261-2

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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