School of Mathematics Colloquium
On the dynamics of holomorphic correspondences on the 2-sphere
by Prof. Gautam Bharali (Indian Institute of Science, Bangalore)
Thursday, July 5, 2012
from
to
(Asia/Kolkata)
at Colaba Campus ( AG-69 )
at Colaba Campus ( AG-69 )
Description |
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 results, seem to depend on whether or not F admits a repeller. We shall discuss what this means, and examine some aspects of the proof of the relevant equidistribution theorem. This is joint work with Shrihari Sridharan. |