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 $f$.

Linear fractional recurrences are given as $z_{n+k}=A\left(z\right)/B\left(z\right)$, where $A\left(z\right)$ and $B\left(z\right)$ are linear functions of $z_n,z_{n+1},\cdots ,z_{n+k-1}$. In this article we consider two questions about these recurrences: (1) Find $A\left(z\right)$ and $B\left(z\right)$ such that the recurrence is periodic; and (2) Find (invariant) integrals in case the induced birational map has quadratic degree growth. We approach these questions by considering the induced birational map and determining its dynamical degree. The first theorem shows that for each $k$ there are $k$-step linear fractional recurrences...

This is a survey about local holomorphic dynamics, from Poincaré's times to nowadays. Some new ideas on how to relate discrete dynamics to continuous dynamics are also introduced. It is the text of the talk given by the author at the XVII UMI Congress at Milano.