Rigidity of some abelian-by-cyclic solvable group actions on $mathbb T^N$

In this paper, we study a natural class of groups that act as affine transformations of $mathbb T^N$. We investigate whether these solvable, abelian-by-cyclic, groups can act smoothly and nonaffinely on $mathbb T^N$ while remaining homotopic to the affine actions. In the affine actions, elliptic and hyperbolic dynamics coexist, forcing a priori complicated dynamics in nonaffine perturbations. We first show, using the KAM method, that any small and sufficiently smooth perturbation of such an affine action can be conjugated smoothly to an affine action, provided certain Diophantine conditions on the action are met. In dimension two, under natural dynamical hypotheses, we get a complete classification of such actions; namely, any such group action by $C^r$ diffeomorphims can be conjugated to the affine action by $C^{r-epsilon}$ conjugacy. Next, we show that in any dimension, $C^1$ small perturbations can be conjugated to an affine action via $C^{1+epsilon}$ conjugacy. The method is a generalization of the Herman theory for circle diffeomorphisms to higher dimensions in the presence of a foliation structure provided by the hyperbolic dynamics.

Projective cocycles over SL(2,R) actions: measures invariant under the upper triangular group

We consider the action of $SL(2,mathbb{R})$ on a vector bundle $mathbf{H}$ preserving an ergodic probability measure $ u$ on the base $X$. Under an irreducibility assumption on this action, we prove that if $hat u$ is any lift of $ u$ to a probability measure on the projectivized bunde $mathbb{P}(mathbf{H})$ that is invariant under the upper triangular subgroup, then $hat u$ is supported in the projectivization $mathbb{P}(mathbf{E}_1)$ of the top Lyapunov subspace of the positive diagonal semigroup. We derive two applications. First, the Lyapunov exponents for the Kontsevich-Zorich cocycle depend continuously on affine measures, answering a question in [MMY]. Second, if $mathbb{P}(mathbf{V})$ is an irreducible, flat projective bundle over a compact hyperbolic surface $Sigma$, with hyperbolic foliation $mathcal{F}$ tangent to the flat connection, then the foliated horocycle flow on $T^1mathcal{F}$ is uniquely ergodic if the top Lyapunov exponent of the foliated geodesic flow is simple. This generalizes results in [BG] to arbitrary dimension.

The C1 generic diffeomorphism has trivial centralizer

Answering a question of Smale, we prove that the space of C1 diffeomorphisms of a compact manifold contains a residual subset of diffeomorphisms whose centralizers are trivial.

C1-generic conservative diffeomorphisms have trivial centralizer

We prove that the spaces of C1 symplectomorphisms and of C1 volume-preserving diffeomorphisms of connected manifolds both contain residual subsets of diffeomorphisms whose centralizers are trivial. (Les diffeomorphismes conservatifs C1-generiques ont un centralisateur trivial. Nous montrons que lespace des symplectomorphismes de classe C1 et lespace des diffeomomorphismes de classe C1 preservant une forme volume contiennent tous deux des sous-ensembles residuels de diffeomorphismes dont le centralisateur est trivial.)