Value distribution of the sequences of the derivatives of iterated polynomials

TL;DR

This paper proves the equidistribution of the derivatives of polynomial iterations' sequences towards the equilibrium measure, with exponential convergence rates, including a parameter space analog for unicritical polynomials.

Contribution

It establishes the equidistribution and exponential convergence of derivatives of polynomial iterates towards the equilibrium measure, extending results to parameter spaces.

Findings

Sequence of averaged pullbacks converges to the equilibrium measure.

Convergence is exponentially fast for most values.

Parameter space analog for unicritical polynomials is provided.

Abstract

We establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any value in $\mathbb{C}\setminus\{0\}$ under the derivatives of the iterations of a polynomials $f\in\mathbb{C}[z]$ of degree more than one towards the $f$-equilibrium (or canonical) measure $\mu_f$ on $\mathbb{P}^1$. We also show that for every $C^2$ test function on $\mathbb{P}^1$, the convergence is exponentially fast up to a polar subset of exceptional values in $\mathbb{C}$. A parameter space analog of the latter quantitative result for the monic and centered unicritical polynomials family is also established.