On stable transitivity of finitely generated groups of volume-preserving diffeomorphisms

Abstract

In this paper, we provide a new criterion for the stable transitivity of volume-preserving finite generated groups on any compact Riemannian manifold. As one of our applications, we generalize a result of Dolgopyat and Krikorian [On simultaneous linearization of diffeomorphisms of the sphere.Duke Math. J. 136(2007), 475–505] and obtain stable transitivity for random rotations on the sphere in any dimension. As another application, we show that for$\infty \geq r\geq 2$, for any$C^{r}$volume-preserving partially hyperbolic diffeomorphism$g$on any compact Riemannian manifold$M$having sufficiently Hölder stable or unstable distribution, for any sufficiently large integer$K$and for any$(f_{i})_{i=1}^{K}$in a$C^{1}$open$C^{r}$dense subset of$\text{Diff}^{r}(M,m)^{K}$, the group generated by$g,f_{1},\ldots ,f_{K}$acts transitively.