Distribution of postcritically finite polynomials iii: Combinatorial continuity

TL;DR

This paper studies the distribution of special polynomial parameters, proving equidistribution results, constructing bifurcation measures for anti-holomorphic families, and exploring combinatorial impressions in moduli spaces.

Contribution

It extends the understanding of parameter distribution in polynomial moduli spaces, introduces a bifurcation measure for quadratic anti-holomorphic families, and exhibits non-trivial combinatorial impressions in degree 4.

Findings

Misiurewicz parameters equidistribute toward the bifurcation measure.

A bifurcation measure for the quadratic anti-holomorphic family is constructed.

Non-trivial impressions of combinatorics are exhibited in degree 4 polynomials.

Abstract

In the first part of the present paper, we continue our study of distribution of postcritically finite parameters in the moduli space of polynomials: we show the equidistribution of Misiurewicz parameters with prescribed combinatorics toward the bifurcation measure. Our results essentially rely on a combinatorial description of the escape locus and of the bifurcation measure developped 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 by Misiurewicz parameters in the spirit of the previous one. Finally, we answer a question of Kiwi, exhibiting in the moduli space of degree 4 polynomials, non-trivial Impression of specific combinatorics.