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
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.
[a]: Division of Mathematics, Kyoto Institute of Technology Sakyo-ku, Kyoto 606-8585 Japan

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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