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A REFORMULATION OF THE DYNAMICAL MANIN–MUMFORD CONJECTURE
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 103 / Issue 1 / February 2021
- Published online by Cambridge University Press:
- 01 June 2020, pp. 154-161
- Print publication:
- February 2021
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- Article
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We advance a new conjecture in the spirit of the dynamical Manin–Mumford conjecture. We show that our conjecture holds for all polarisable endomorphisms of abelian varieties and for all polarisable endomorphisms of
$(\mathbb{P}^{1})^{N}$. Furthermore, we show various examples which highlight the restrictions one would need to consider in formulating any general conclusion in the dynamical Manin–Mumford conjecture.
The dynamical Manin–Mumford conjecture and the dynamical Bogomolov conjecture for endomorphisms of
$(\mathbb{P}^{1})^{n}$
- Part of
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- Journal:
- Compositio Mathematica / Volume 154 / Issue 7 / July 2018
- Published online by Cambridge University Press:
- 21 May 2018, pp. 1441-1472
- Print publication:
- July 2018
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- Article
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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.
