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 $d \geq 6$ with rational coefficients. We also investigate the quantity $\liminf_{x \in \overline{\mathbb{Q}}} \left(\widehat{h}_f(x) + \widehat{h}_g(x) \right)$ for a generic pair of polynomials and prove both lower and upper bounds for it.