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Dynamics on ℙ1: preperiodic points and pairwise stability

Part of: Arithmetic and non-Archimedean dynamical systems Complex dynamical systems Arithmetic algebraic geometry

Published online by Cambridge University Press:  05 January 2024

Laura DeMarco  and
Niki Myrto Mavraki
Laura DeMarco
Affiliation:
Department of Mathematics, Harvard University, Science Center Room 325, 1 Oxford Street, Cambridge, MA 02138, USA demarco@math.harvard.edu
Niki Myrto Mavraki
Affiliation:
Department of Mathematics, University of Toronto, Bahen Centre, 40 St. George Street, Toronto, Ontario, M5S 2E4, Canada myrto.mavraki@utoronto.ca
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Abstract

DeMarco, Krieger, and Ye conjectured that there is a uniform bound B, depending only on the degree d, so that any pair of holomorphic maps $f, g :{\mathbb {P}}^1\to {\mathbb {P}}^1$ with degree $d$ will either share all of their preperiodic points or have at most $B$ in common. Here we show that this uniform bound holds for a Zariski open and dense set in the space of all pairs, $\mathrm {Rat}_d \times \mathrm {Rat}_d$, for each degree $d\geq 2$. The proof involves a combination of arithmetic intersection theory and complex-dynamical results, especially as developed recently by Gauthier and Vigny, Yuan and Zhang, and Mavraki and Schmidt. In addition, we present alternate proofs of the main results of DeMarco, Krieger, and Ye [Uniform Manin-Mumford for a family of genus 2 curves, Ann. of Math. (2) 191 (2020), 949–1001; Common preperiodic points for quadratic polynomials, J. Mod. Dyn. 18 (2022), 363–413] and of Poineau [Dynamique analytique sur $\mathbb {Z}$ II : Écart uniforme entre Lattès et conjecture de Bogomolov-Fu-Tschinkel, Preprint (2022), arXiv:2207.01574 [math.NT]]. In fact, we prove a generalization of a conjecture of Bogomolov, Fu, and Tschinkel in a mixed setting of dynamical systems and elliptic curves.

Keywords

equidistributionheightscurrentsbifurcationrational functionspreperiodic points

MSC classification

Primary: 37F10: Polynomials; rational maps; entire and meromorphic functions 37P15: Global ground fields
Secondary: 11G50: Heights

Information

Type
Research Article
Information
Compositio Mathematica , Volume 160 , Issue 2 , February 2024 , pp. 356 - 387
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Copyright
© 2024 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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