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Shared dynamically small points for polynomials on average
- Part of
-
- Journal:
- Ergodic Theory and Dynamical Systems , First View
- Published online by Cambridge University Press:
- 01 September 2025, pp. 1-44
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- Article
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Given two rational maps
$f,g: \mathbb {P}^1 \to \mathbb {P}^1$ of degree d over 
$\mathbb {C}$, DeMarco, Krieger, and Ye [Common preperiodic points for quadratic polynomials. J. Mod. Dyn. 18 (2022), 363–413] have 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 
$\liminf _{x \in \overline {\mathbb {Q}}} (\widehat {h}_f(x) + \widehat {h}_g(x) )$ for a generic pair of polynomials and prove both lower and upper bounds for it.
