Equidistribution, multipliers, and moving targets in polynomial families

Source abstract

We study equidistribution of parameters for each of which a polynomial has a cycle with a prescribed moving multiplier in the full critically marked or unicritical polynomial families of a given degree $d>1$. For the full critically marked monic polynomial family $f=(f_λ)$, $λ\in\mathbb{C}^{d-1}$, of degree $d$, letting $p_{f,n}^*(λ,w)$ be the multiplier polynomial for cycles of formal exact period $n$ having multiplier $w$, we show that for any target sequence $w_n=d^na_n$ satisfying $n^{-1}\log|a_n|\to\ell\in[-\infty,+\infty)$, the averaged current of integration $ν(n)^{-1}[p_{f,n}^*(\cdot,w_n)=0]$ along the zero divisor of $p_{f,n}^*(\cdot,w_n)$ tends to the $(1,1)$-current $\mathrm{d}\mathrm{d}^c\max\{L_d,\log d+\ell\}$ in the sense of currents on $\mathbb{C}^{d-1}$ as $n\to\infty$, where $L_d$ is the Lyapunov exponent function and $ν(n)$ is the degree of the $n$th formal exact dynatomic polynomial associated to $f$. We also obtain the corresponding equidistribution theorem for the unicritical family $f_λ(z)=z^d+λ$, $λ\in\mathbb{C}$, of degree $d$. The proof combines thermodynamic formalism, the Eremenko--Levin multiplier bound, and a comparison principle for plurisubharmonic functions, in a partly quantitative manner. A key ingredient is the vanishing of the bifurcation measure on the boundary of every component of the interior of the connectedness locus for the full critically marked family, which we show using Kiwi's study of critical portraits and rational laminations.