Common preperiodic points for quadratic polynomials

2019/11/06 by DeMarco, Laura, Krieger, Holly, Ye, Hexi · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1911.02458

Abstract

Let f_c(z) = z²+c for c ∈ ℂ. We show there exists a uniform bound on the number of points in ℙ¹(ℂ) that can be preperiodic for both f_c1 and f_c2 with c₁\not= c₂ in ℂ. The proof combines arithmetic ingredients with complex-analytic; we estimate an adelic energy pairing when the parameters lie in ℚ, building on the quantitative arithmetic equidistribution theorem of Favre and Rivera-Letelier, and we use distortion theorems in complex analysis to control the size of the intersection of distinct Julia sets. The proof is effective, and we provide explicit constants for each of the results.

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