We study the dynamics of holomorphic correspondences f on a compact Riemann surface X in the case, so far not well understood, where f and f −1 have the same topological degree. Under a mild and necessary condition that we call non weak modularity, f admits two canonical probability measures µ + and µ − which are invariant by f * and f * respectively. If the critical values of f (resp. f −1 ) are not periodic, the backward (resp. forward) orbit of any point a ∈ X equidistributes towards µ + (resp. µ − ), uniformly in a and exponentially fast.

Let µ be a probability measure on GL d (R) and denote by S n := g n • • • g 1 the associated random matrix product, where g j are i.i.d. with law µ. Under the assumptions that µ has a finite exponential moment and generates a proximal and strongly irreducible semigroup, we prove a Berry-Esseen bound with the optimal rate O(1/√ n) and a generalLocal Limit Theorem for the coefficients of S n .

Let X be a compact Kähler manifold and let T be a foliated cycle directed by a transversally Lipschitz lamination on X . We prove that the self-intersection of the cohomology class of T vanishes as long as T does not contain currents of integration along compact manifolds.As a consequence we prove that Lipschitz laminations of low codimension in certain manifolds, e.g. projective spaces, do not carry any foliated cycles except those given by integration along compact leaves.

No abstract