Nevanlinna theory and value distribution in the unicritical polynomials family

TL;DR

This paper uses Nevanlinna theory and complex dynamics to analyze the distribution of parameters in unicritical polynomial families, establishing quantitative equidistribution results towards the bifurcation current with explicit error estimates.

Contribution

It introduces a novel approach combining Nevanlinna theory with complex dynamics to quantify parameter distribution in unicritical polynomials.

Findings

Quantitative equidistribution towards the bifurcation current with error estimates

Distribution of parameters with superattracting periodic points analyzed

Application of Nevanlinna theory in complex dynamics context

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.