Riv. Mat. Univ. Parma, to appear

Yûsuke Okuyama [a]

Nevanlinna theory and value distribution in the unicritical polynomials family

Pages:
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.