Regularity of conjugacies of algebraic actions of Zariski-dense groups

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 SLd (Z) acting on the torus T-d and Zariski-dense subgroups of a simple non-compact Lie group G acting on a compact homogeneous space X of G with an invariant measure.