Equidistribution, multipliers, and moving targets in polynomial families
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=(fλ), λ∈ℂd−1, of degree d, letting pf,n*(λ,w) be the multiplier polynomial for cycles of formal exact period n having multiplier w, we show that for any target sequence wn=dnan satisfying n−1log|an|→ℓ∈[−∞,+∞), the averaged current of integration ν(n)−1[pf,n*(·,wn)=0] along the zero divisor of pf,n*(·,wn) tends to the (1,1)-current ddcmax{Ld,log d+ℓ} in the sense of currents on ℂd−1 as n→∞, where Ld is the Lyapunov exponent function and ν(n) is the degree of the nth formal exact dynatomic polynomial associated to f. We also obtain the corresponding equidistribution theorem for the unicritical family fλ(z)=zd+λ, λ∈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.
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BibTeX
@misc{okuyama2026equidistribution,
title = {{Equidistribution, multipliers, and moving targets in polynomial families}},
author = {Yûsuke Okuyama},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25509},
eprint = {2610.25509},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25509},
}
RIS
TY - UNPB
TI - Equidistribution, multipliers, and moving targets in polynomial families
AU - Yûsuke Okuyama
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25509
UR - https://arxiv.org/abs/2610.25509
AB - 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∈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^−1log|a_n|→ℓ∈[−∞,+∞), the averaged current of integration ν(n)^−1[p_f,n^*(·,w_n)=0] along the zero divisor of p_f,n^*(·,w_n) tends to the (1,1)-current dd^cmax{L_d,log d+ℓ} in the sense of currents on ℂ^d−1 as n→∞, where L_d is the Lyapunov exponent function and ν(n) is the degree of the nth 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∈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.
ER -