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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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$p$-adic interpolation of orbits under rational maps
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by Jason P. Bell and Xiao Zhong
Proc. Amer. Math. Soc. 151 (2023), 4661-4672
DOI: https://doi.org/10.1090/proc/16591
Published electronically: August 18, 2023

Abstract:

Let $L$ be a field of characteristic zero, let $h:\mathbb {P}^1\to \mathbb {P}^1$ be a rational map defined over $L$, and let $c\in \mathbb {P}^1(L)$. We show that there exists a finitely generated subfield $K$ of $L$ over which both $c$ and $h$ are defined along with an infinite set of inequivalent non-archimedean completions $K_{\mathfrak {p}}$ for which there exists a positive integer $a=a(\mathfrak {p})$ with the property that for $i\in \{0,\ldots ,a-1\}$ there exists a power series $g_i(t)\in K_{\mathfrak {p}}[[t]]$ that converges on the closed unit disc of $K_{\mathfrak {p}}$ such that $h^{an+i}(c)=g_i(n)$ for all sufficiently large $n$. As a consequence we show that the dynamical Mordell-Lang conjecture holds for split self-maps $(h,g)$ of $\mathbb {P}^1 \times X$ with $g$ an étale self-map of a quasiprojective variety $X$.
References
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Bibliographic Information
  • Jason P. Bell
  • Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
  • MR Author ID: 632303
  • ORCID: 0000-0002-1483-9769
  • Email: jpbell@uwaterloo.ca
  • Xiao Zhong
  • Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
  • Email: x48zhong@uwaterloo.ca
  • Received by editor(s): February 3, 2022
  • Received by editor(s) in revised form: August 28, 2022, and November 16, 2022
  • Published electronically: August 18, 2023
  • Additional Notes: The authors were supported by NSERC grant RGPIN-2016-03632.
  • Communicated by: Rachel Pries
  • © Copyright 2023 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 151 (2023), 4661-4672
  • MSC (2020): Primary 37F10, 37P20, 37P55
  • DOI: https://doi.org/10.1090/proc/16591
  • MathSciNet review: 4634872