Riv. Mat. Univ. Parma, Vol. 9, No. 1, 2018

Yûsuke Okuyama [a]

Nevanlinna theory and value distribution in the unicritical polynomials family

Pages: 1-19
Received: 26 April 2017
Accepted in revised form: 14 March 2018
Mathematics Subject Classification (2010): Primary 37F45; Secondary 30D35.
Keywords: Unicritical polynomials family, superattracting periodic point, equidistribution, Nevanlinna theory.
Author address:
[a]: Division of Mathematics, Kyoto Institute of Technology Sakyo-ku, Kyoto 606-8585 Japan

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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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