Zhiyuan Zhang: On stable transitivity of finitely generated group of volume preserving diffeomorphisms

It is conjectured that for any m \geq 2, the group
generated by m generic C^r volume preserving diffeomorphisms on a
compact manifold M, acts transitively on M. We give a criterion for
the stable transitivity of the action of finitely generated group of
volume preserving diffeomorphisms on any compact manifold. As an
application, we generalise a result of Dolgopyat and Krikorian, and
obtained stable transitivity for random rotations on the sphere in any
dimension. As another application, we showed that for any C^{\infty}
volume preserving partially hyperbolic diffeomorphism g on any compact
Riemannian manifold M having sufficiently Holder stable or unstable
distribution, for any sufficiently large integer K, for any K
diffeomorphisms f_1, ... , f_K in a C^1 open C^{\infty} dense subset
of Diff^r(M,Vol)^K, the group generated by g, f_1,..., f_K is
transitive.​