Conformal measures and matings between Kleinian groups and quadratic polynomials
Volume 193 / 2007
Fundamenta Mathematicae 193 (2007), 95-132
MSC: 37F35, 37F30, 37F10, 37F05, 37F45.
DOI: 10.4064/fm193-2-1
Abstract
Following results of McMullen concerning rational maps, we show that the limit set of matings between a certain class of representations of $C_2 \,\ast\, C_3$ and quadratic polynomials carries $\delta$-conformal measures, and that if the correspondence is geometrically finite then the real number $\delta$ is equal to the Hausdorff dimension of the limit set. Moreover, when $f$ is the limit of a pinching deformation $\{f_t\}_{0 \leq t < 1}$ we give sufficient conditions for the dynamical convergence of $\{f_t\}$.
