Nevanlinna theory and value distribution in the unicritical polynomials family

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.

arXiv comment14 pages. (v2) A few words are corrected