On stable transitivity of finitely generated groups of volume preserving diffeomorphisms

In this paper, we provide a new criterion for the stable transitivity of volume preserving finite generated group on any compact Riemannian manifold. As one of our applications, we generalised a result of Dolgopyat and Krikorian in \cite{DK} and obtained stable transitivity for random rotations on the sphere in any dimension. As another application, we showed that for $\infty \geq r \geq 2$, any $C^r$ volume preserving partially hyperbolic diffeomorphism $g$ on any compact Riemannian manifold $M$ having sufficiently H\"older stable or unstable distribution, for any sufficiently large integer $K$, for any $(f_i)_{i=1}^{K}$ in a $C^1$ open $C^r$ dense subset of $\textnormal{\Diff}^r(M,m)^K$, the group generated by $g, f_1,\cdots, f_K$ acts transitively.