The dynamics of holomorphic correspondences on Riemann surfaces

Abstract : Owing to the theorem of M. de Franchis, there cannot be any interesting classical holomorphic dynamics on a compact hyperbolic Riemann surface. However, these surfaces are rich in holomorphic correspondences. The iteration of such a correspondence gives rise to many questions -- which have only recently begun to be investigated -- that are analogous to those in the theory of iterations of rational maps on the 2-sphere. A part of this talk will be devoted to defining the analogue of the Fatou set for correspondences.

Let F be a holomorphic correspondence on a compact Riemann surface X. Under certain conditions, F admits a canonical invariant measure \mu_F with good ergodic properties. The work of Dinh and Sibony shows that, for a generic point p in X, the normalized sums of point masses carried by the pre-images of p under successive iterates of F converge to \mu_F. Extending the theory of iterative dynamics on the 2-sphere, the support of \mu_F is a good candidate for the analogue of the Julia set. A basic question then is: do this and (the analogue of) the Fatou set partition X? It turns out that the answer in general is, "No." We shall discuss some results and a conjecture towards producing a good partition of X that would -- for correspondences -- serve as the analogue of the classical Fatou-Julia dichotomy.