### 18 december 2008, kl 13.15-14.15, KTH, sal 3721.

###
Juan Rivera-Letelier (PUC, Chile): Ergodic theory of ultrametric rational maps

**Abstract.**
One the fundamental results in complex dynamics is the existence
and uniqueness of the measure of maximal entropy. This measure has several
interesting properties: it has exponential decay of correlations and
satisfies the central limit theorem for Holder continuous observables, and
furthermore it describes the asymptotic distribution of periodic points
and of iterated preimages. In the arithmetic case it also describes the
asymptotic distribution of points of small height.
In the case of rational maps defined over an ultrametric field there is a
measure with similar equidistribution properties. However some relatively
simple examples show that this measure is not of maximal entropy, and that
the topological entropy is not equal to the logarithm of the degree of the
rational map. There are however some estimates that allow us to
characterize those rational maps with zero topological entropy.
The purpose of this talk is to describe the construction of this measure,
and to show some of its fundamental properties. We will also see some key
examples and open problems.