Description du sujet de thèse

Dynamics and bifurcations of holomorphic families of correspondences. Holomorphic correspondences are a class of complex dynamical systems that generalize both rational maps and finitely generated Kleinian groups. The goal of this project is to develop a bifurcation theory for these systems using pluripotential theory.

A well-established framework in holomorphic dynamics concerns families of rational maps: the work of Mañé-Sad-Sullivan and DeMarco has shown that the Julia set moves holomorphically with the parameter if and only if the Lyapunov exponent of the equilibrium measure is a pluriharmonic function, which is equivalent to the stability of periodic orbits. Similar results have been obtained by Deroin and Dujardin for Kleinian groups. This project aims to generalize these results to holomorphic correspondences, for which Dinh and his coauthors have already established the existence and uniqueness of an equilibrium measure in certain classes.

The proposed approach relies on several tools from pluripotential theory, ergodic theory, and complex analysis. The objective is to define and study the Lyapunov exponent associated with the equilibrium measure and to examine the relationship between the support of dd^c L and the dependence of the Julia set on the parameters.

Contexte de travail

The recruited person will prepare a doctoral thesis in mathematics in the laboratory Denis Poisson Institute in Orléans. They will conduct research under the supervision of Matthieu ASTORG and Lucas KAUFMANN, actively participate in the laboratory seminars, as well as in national and international conferences. They will also contribute to the scientific valorization of their results by writing and publishing research articles. They will work on dynamical questions related to bifurcations of families of holomorphic correspondences. They will join the ANG team (Analysis and Geometry) and will be associated with the various research activities of the laboratory. Missions may potentially be planned in France or abroad (summer schools, seminars, or international conferences).

Contraintes et risques

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