| 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.
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