Large deviation of preimages of points under the dynamics of a holomorphic correspondence

Abstract

Our main objective in this paper is to prove a large deviation result for the distribution of preimages of a point under the iteration of a holomorphic correspondence with respect to the unique equilibrium state associated to a Hölder continuous potential defined on the support of the Dinh-Sibony measure of the correspondence. As a consequence, we obtain a convergence result which generalises the result on equidistribution of preimages by Dinh and Sibony. We also study a situation where the Dinh-Sibony measure happens to be the unique measure of maximal entropy for an appropriate holomorphic correspondence and also record some comments that connect this work with some recent works in the literature.