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In the space $\mathbb{C}$ of the parameters $\lambda$ of the unicriticalpolynomials family $f(\lambda,z)=f_\lambda(z)=z^d+\lambda$ of degree $d>1$, weestablish a quantitative equidistribution result towards the bifurcationcurrent (indeed measure) $T_f$ of $f$ as $n\to\infty$ on the averageddistributions of all parameters $\lambda$ such that $f_\lambda$ has asuperattracting periodic point of period $n$ in $\mathbb{C}$, with a concreteerror estimate for $C^2$-test functions on $\mathbb{P}^1$. In the proof, notonly complex dynamics but also a standard argument from the Nevanlinna theoryplay key roles.

Nevanlinna theory and value distribution in the unicritical polynomials family