Value distribution of the sequences of the derivatives of iterated polynomials

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.

12 pages. (v3) corrected a few typos; (v2) Theorem 3 now focuses on a parameter space analog of Theorem 2 for the unicritical polynomials family