Distribution of postcritically finite polynomials II: Speed of convergence

TL;DR

This paper proves that postcritically finite polynomials in the moduli space equidistribute toward the bifurcation measure at an exponential rate, using complex analytic and pluripotential theory methods, improving previous arithmetic results.

Contribution

It establishes the exponential speed of convergence for equidistribution of postcritically finite polynomials toward the bifurcation measure using complex analytic techniques.

Findings

Exponential speed of convergence for C^2-observables.

Equidistribution of hyperbolic parameters with given multipliers.

Explicitly describes parameters with prescribed multipliers outside a pluripolar set.

Abstract

In the moduli space of degree d polynomials, we prove the equidistribution of postcritically finite polynomials toward the bifurcation measure. More precisely, using complex analytic arguments and pluripotential theory, we prove the exponential speed of convergence for C 2-observables. This improves results obtained with arithmetic methods by Favre and Rivera-Letellier in the unicritical family and Favre and the first author in the space of degree d polynomials. We deduce from that the equidistribution of hyperbolic parameters with (d -- 1) distinct attracting cycles of given multipliers toward the bifurcation measure with exponential speed for C 1-observables. As an application, we prove the equidistribution (up to an explicit extraction) of parameters with (d -- 1) distinct cycles with prescribed multiplier toward the bifurcation measure for any (d -- 1) multipliers outside a pluripolar set.