Colloquium abstracts

Gautam Bharali
Indian Institute of Science, Bangalore
July 5, 2012

On the dynamics of holomorphic correspondences on the 2-sphere:  We shall look at a couple of equidistribution results for holomorphic correspondences on the 2-sphere. Our results have the following character: if $F$ is a holomorphic correspondence on the 2-sphere, then, under certain conditions, $F$ admits an equilibrium measure $\mu$, and, for a generic point $p$ in the sphere, the normalized sums of point masses carried by the pre-images of $p$ under successive iterates of $F$ converge to $\mu$. Now, let $F^t$ denote the transpose of $F$. Under the condition $d_{top}(F) > d_{top}(F^t)$, where $d_{top}$ denotes the topological degree, our result is a small refinement of a set of recent results by Dinh and Sibony. However, for most interesting correspondences on the 2-sphere, $d_{top}(F) \leq d_{top}(F^t)$. This is certainly the case for the correspondences introduced by Bullett and Penrose --- who were among the first to introduce these objects. When $d_{top}(F) \leq d_{top}(F^t)$, the existence of equilibrium measures, and equidistribution resul