Nevanlinna theory and value distribution in the unicritical polynomials family

Yûsuke Okuyama

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arXiv · 1802.02723

Nevanlinna theory and value distribution in the unicritical polynomials family

Yûsuke Okuyama

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Abstract

In the space $\mathbb{C}$ of the parameters $\lambda$ of the unicritical polynomials family $f(\lambda,z)=f_\lambda(z)=z^d+\lambda$ of degree $d>1$, we establish a quantitative equidistribution result towards the bifurcation current (indeed measure) $T_f$ of $f$ as $n\to\infty$ on the averaged distributions of all parameters $\lambda$ such that $f_\lambda$ has a superattracting periodic point of period $n$ in $\mathbb{C}$, with a concrete error estimate for $C^2$-test functions on $\mathbb{P}^1$. In the proof, not only complex dynamics but also a standard argument from the Nevanlinna theory play key roles.

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BibTeX RIS

Yûsuke Okuyama. 2018-02-08. Nevanlinna theory and value distribution in the unicritical polynomials family. https://arxiv.org/abs/1802.02723

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