Value distribution of the sequences of the derivatives of iterated polynomials

15 Nov 2016  ·  Okuyama Yûsuke ·

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. read more

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Complex Variables Dynamical Systems