0000000000611234
AUTHOR
Sylvain Crovisier
showing 3 related works from this author
Tame dynamics and robust transitivity
2011
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 C1-open set U of tame diffeomorphisms (their dynamics only splits into finitely many chain-recurrence classes) such that for any diffeomorphism in a C-infinity-dense subset of U, one of the chain-recurrence classes is not transitive (and has an isolated point). Moreover, these dynamics are obtained…
Global dominated splittings and the C1 Newhouse phenomenom
2006
International audience; We prove that given a compact n-dimensional boundaryless manifold M, n >=2, there exists a residual subset R of the space of C1 diffeomorphisms Diff such that given any chain-transitive set K of f in R then either K admits a dominated splitting or else K is contained in the closure of an infinite number of periodic sinks/sources. This result generalizes the generic dichotomy for homoclinic classes in [BDP]. It follows from the above result that given a C1-generic diffeomorphism f then either the nonwandering set Omega(f) may be decomposed into a finite number of pairwise disjoint compact sets each of which admits a dominated splitting, or else f exhibits infinitely m…
Centralizers of C^1-generic diffeomorphisms
2006
On the one hand, we prove that the spaces of C^1 symplectomorphisms and of C^1 volume-preserving diffeomorphisms both contain residual subsets of diffeomorphisms whose centralizers are trivial. On the other hand, we show that the space of C^1 diffeomorphisms of the circle and a non-empty open set of C^1 diffeomorphisms of the two-sphere contain dense subsets of diffeomorphisms whose centralizer has a sub-group isomorphic to R.