There is a remarkable equidistribution phenomena of complex rational maps: the iterated preimages of every (non-exceptional) point are equidistributed according to a certain measure, often called equilibrium measure (Brolin, Lyubich, Freire-Lopez-Mañé).

On the other hand, more recently a series of equidistribution of small points phenomena have been discovered, that have several geometric and arithmetic implications. For example, in an Abelian variety defined over a number field, the points of small Neron-Tate height are equidistributed with respect to the normalized Haar measure (Szpiro-Ullmo-Zhang).

We will prove a result in the intersection of these topics: given a rational map with algebraic coefficients, the points of small canonical height are equidistributed with respect to the equilibrium measure. Here the equisitribution occurs at all places i.e. in the complex and in the p-adic setting. As a corollary we obtain an arithmetic proof of Brolin equidistribution result, mentioned above.

The proof is based on potential theory in the complex plane and in Berkovich spaces and in a remarkable fact: the canonical height can be written as a sum of local energies.

This conference is based on the following joint work with C. Favre (CNRS-Paris VII):
Théorème d'équidistribution de Brolin en dynamique p-adique.
Appeared in Comptes Rendus Acad. Sci. Paris, 2004
http://arxiv.org/math.DS/0407469
Equidistribution des points de petite hauteur.
http://arxiv.org/math.NT/0407471