Distribution of postcritically finite polynomials III: Combinatorial continuity
In the first part of the present paper, we continue our study of the distribution of postcritically finite parameters in the moduli space of polynomials: we show the equidistribution of PCF Misiurewicz parameters with prescribed combinatorics with respect to the bifurcation measure. Our results essentially rely on a combinatorial description of the escape locus and of the bifurcation measure developed by Kiwi and Dujardin–Favre.
In the second part of the paper, we construct a bifurcation measure for the connectedness locus of the quadratic anti-holomorphic family which is supported by a strict subset of the boundary of the Tricorn. We also establish an approximation property of PCF Misiurewicz parameters in the spirit of the previous one.