Soit $f$ un endomorphisme holomorphe de ${\mathbb{P}}^k\left(ℂ\right)$. Je présenterai une construction géométrique, due à Briend et Duval, d’une mesure de probabilité $\mu$ ayant les propriétés suivantes : $\mu$ reflète la distribution des préimages des points en dehors d’un ensemble exceptionnel algébrique, les points périodiques répulsifs de $f$ s’équidistribuent par rapport à $\mu$ et $\mu$ est l’unique mesure d’entropie maximale de $f$.

We consider measure-preserving diffeomorphisms of the torus with zero entropy. We prove that every ergodic $C^1$-diffeomorphism with linear growth of the derivative is algebraically conjugate to a skew product of an irrational rotation on the circle and a circle $C^1$-cocycle. We also show that for no positive β ≠ 1 does there exist an ergodic $C^2$-diffeomorphism whose derivative has polynomial growth with degree β.

We consider $C^2$ families $t\mapsto f_t$ of $C^4$ unimodal maps $f_t$ whose critical point is slowly recurrent, and we show that the unique absolutely continuous invariant measure ${\mu }_t$ of $f_t$ depends differentiably on $t$, as a distribution of order $1$. The proof uses transfer operators on towers whose level boundaries are mollified via smooth cutoff functions, in order to avoid artificial discontinuities. We give a new representation of ${\mu }_t$ for a Benedicks-Carleson map $f_t$, in terms of a single smooth function and the inverse branches...