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Shared dynamically small points for polynomials on average

Part of: Arithmetic and non-Archimedean dynamical systems

Published online by Cambridge University Press:  01 September 2025

YAN SHENG ANG  and
JIT WU YAP Open the ORCID record for JIT WU YAP [Opens in a new window]
YAN SHENG ANG
Affiliation:
Department of Mathematics, https://ror.org/042nb2s44 Massachusetts Institute of Technology , Cambridge, MA 02139, USA (e-mail: angyansheng@yahoo.com.sg)
JIT WU YAP*
Affiliation:
Department of Mathematics, https://ror.org/03vek6s52 Harvard University , Cambridge, MA 02138, USA
*
e-mail: jyap@math.harvard.edu
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Abstract

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.

Keywords

preperiodic pointsenergy pairingarithmetic dynamicsunlikely intersection

MSC classification

Primary: 37P05: Polynomial and rational maps 37P30: Height functions; Green functions; invariant measures
Secondary: 37P35: Arithmetic properties of periodic points 37P50: Dynamical systems on Berkovich spaces

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 44
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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