Executive Summary : | The principal objective of this project is to study various topological, analytic, and measure-theoretic properties of conformal dynamical systems that arise as combinations of (anti-)holomorphic polynomials and Kleinian (reflection) groups. On the topological/analytic side, we will investigate degenerations of conformal matings as the (anti-)polynomial tends to the boundary of the hyperbolic component and the Kleinian (reflection) group tends to the boundary of the Teichmüller space. We intend to obtain analogues of Thurston's double limit theorem in these hybrid settings, and analyze special situations where the topology of the domain of definition of the conformal mating undergoes "phase transition". This phase transition is an entirely new phenomenon in the study of degenerations of complex dynamical systems. From the measure-theoretic perspective, we will look at the measures of maximal entropy of the conformal matings, which are related to the so-called Patterson-Sullivan measures of Kleinian groups and measures of maximal entropy of polynomials. We will study how the Hausdorff dimension of this measure and the Hausdorff dimension of the limit set of the conformal mating vary as the polynomial and the group vary over their deformation spaces, and attempt to link the variation of the Hausdorff dimension to suitable pressure metrics on hyperbolic components and Weil-Peterson metric on Teichmüller spaces. This would be a novel result combining metric properties of two different kinds of parameter spaces in a single parameter space. Finally, we will also carry out a multifractal analysis for these measures of maximal entropy and for analogous measures associated with higher Bowen-Series maps which are orbit equivalent to Fuchsian punctured sphere groups. |