Shared Dynamically-Small Points for Polynomials on Average

Abstract

Given two rational maps $f,g:{\mathbb{P}}^1\to {\mathbb{P}}^1$ of degree $d$ over $\mathbb{C}$, DeMarco-Krieger-Ye [DKY22] has conjectured that there should be a uniform bound $B=B(d)>0$ such that either they have at most $B$ common preperiodic points or they have the same set of preperiodic points. We study their conjecture from a statistical perspective and prove that the average number of shared preperiodic points is zero for monic polynomials of degree $d\ge 6$ with rational coefficients. We also investigate the quantity ${\mathrm{liminf}}_{x\in \overline{\mathbb{Q}}}\left({\hat{h}}_f(x)+{\hat{h}}_g(x)\right)$ for a generic pair of polynomials and prove both lower and upper bounds for it.

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