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

Published online by Cambridge University Press:  02 May 2017

ZHIYUAN ZHANG
ZHIYUAN ZHANG*
Affiliation:
Institut de Mathématique de Jussieu—Paris Rive Gauche, Bâtiment Sophie Germain, Bureau 652, 75205 Paris Cedex 13, France email zzzhangzhiyuan@gmail.com
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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.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 2 , February 2019 , pp. 554 - 576
Copyright
© Cambridge University Press, 2017 

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