# American Institute of Mathematical Sciences

July  2008, 2(3): 509-540. doi: 10.3934/jmd.2008.2.509

## Regularity of conjugacies of algebraic actions of Zariski-dense groups

 1 School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom 2 Department of Mathematics, University of Northern Iowa, Cedar Falls, IA 50614-0506, United States 3 Department of Mathematics, 2074 East Hall, 530 Church Street, University of Michigan, Ann Arbor, MI 48109-1043, United States

Received  April 2008 Published  April 2008

Let $\alpha_0$ be an affine action of a discrete group $\Gamma$ on a compact homogeneous space $X$ and $\alpha_1$ a smooth action of $\Gamma$ on $X$ which is $C^1$-close to $\alpha_0$. We show that under some conditions, every topological conjugacy between $\alpha_0$ and $\alpha_1$ is smooth. In particular, our results apply to Zariski-dense subgroups of $SL_d(\mathbb{Z})$ acting on the torus $\mathbb{T}^d$ and Zariski-dense subgroups of a simple noncompact Lie group $G$ acting on a compact homogeneous space $X$ of $G$ with an invariant measure.
Citation: Alexander Gorodnik, Theron Hitchman, Ralf Spatzier. Regularity of conjugacies of algebraic actions of Zariski-dense groups. Journal of Modern Dynamics, 2008, 2 (3) : 509-540. doi: 10.3934/jmd.2008.2.509
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