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The dynamical Manin–Mumford conjecture and the dynamical Bogomolov conjecture for endomorphisms of $(\mathbb{P}^{1})^{n}$

Part of: Arithmetic and non-Archimedean dynamical systems

Published online by Cambridge University Press:  21 May 2018

Dragos Ghioca ,
Khoa D. Nguyen  and
Hexi Ye
Dragos Ghioca
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada email [email protected]
Khoa D. Nguyen
Affiliation:
University of Calgary, Mathematical Sciences Building MS 542, 2500 University Drive NW, Calgary, AB T2N 4T4, Canada email [email protected]
Hexi Ye
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou, 310027, China email [email protected]
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Abstract

We prove Zhang’s dynamical Manin–Mumford conjecture and dynamical Bogomolov conjecture for dominant endomorphisms $\unicode[STIX]{x1D6F7}$ of $(\mathbb{P}^{1})^{n}$ . We use the equidistribution theorem for points of small height with respect to an algebraic dynamical system, combined with an analysis of the symmetries of the Julia set for a rational function.

Keywords

dynamical Manin–Mumford conjectureequidistribution of points of small heightsymmetries of the Julia set of a rational function

MSC classification

Primary: 37P05: Polynomial and rational maps
Secondary: 37P30: Height functions; Green functions; invariant measures
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 7 , July 2018 , pp. 1441 - 1472
Copyright
© The Authors 2018 

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