# Stable accessibility is C1 dense

@inproceedings{Dolgopyat2001StableAI, title={Stable accessibility is C1 dense}, author={Dmitry Dolgopyat and Amie Wilkinson}, year={2001} }

— We prove that in the space of ail Cr (r ^ 1) partially hyperbolic diffeomorphisms, there is a C1 open and dense set of accessible diffeomorphisms. This settles the C1 case of a conjecture of Pugh and Shub. The same resuit holds in the space of volume preserving or symplectic partially hyperbolic diffeomorphisms. Combining this theorem with results in [Br], [Ar] and [PugSh3], we obtain several corollaries. The first states that in the space of volume preserving or symplectic partially…

## 47 Citations

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## References

SHOWING 1-10 OF 22 REFERENCES

An open dense set of stably ergodic diffeomorphisms in a neighborhood of a non-ergodic one

- Mathematics
- 2001

SRB measures for partially hyperbolic systems whose central direction is mostly contracting

- Mathematics
- 2000

We consider partially hyperbolic diffeomorphisms preserving a splitting of the tangent bundle into a strong-unstable subbundleEuu (uniformly expanding) and a subbundleEc, dominated byEuu.We prove…

Stable ergodicity and julienne quasi-conformality

- Mathematics
- 2000

Abstract. In this paper we dramatically expand the domain of known stably ergodic, partially hyperbolic dynamical systems. For example, all partially hyperbolic affine diffeomorphisms of compact…

Stably Ergodic Dynamical Systems and Partial Hyperbolicity

- MathematicsJ. Complex.
- 1997

It is shown that a little hyperbolicity goes a long way toward guaranteeing stable ergodicity, and in fact may be necessary for it, and examples to which this theory applies include translations on certain homogeneous spaces and the time-one map of the geodesic flow for a manifold of constant negative curvature.

Controllability of Nonlinear Systems on Compact Manifolds

- Mathematics
- 1974

It is proved that a nonlinear conservative control system on a compact manifold is controllable if (and only if in the analytical case) a certain condition expressed in terms of the “Taylor expansion…

Stably ergodic approximation: two examples, Ergod

- Th. and Dyam. Syst
- 2000

The generic sympletic C1 diffeomorphisms of 4-dimensional sympletic mamfolds are hyperbolic, partially hyperbolic or have a completely elliptic periodic point, Prépublications d'Orsay 2000-17

- 2000