Algebraic dynamics of skew-linear self-maps
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- by Dragos Ghioca and Junyi Xie PDF
- Proc. Amer. Math. Soc. 146 (2018), 4369-4387 Request permission
Abstract:
Let $X$ be a variety defined over an algebraically closed field $k$ of characteristic $0$, let $N\in \mathbb {N}$, let $g:X\dashrightarrow X$ be a dominant rational self-map, and let $A:\mathbb {A}^N\longrightarrow \mathbb {A}^N$ be a linear transformation defined over $k(X)$, i.e., for a Zariski open dense subset $U\subset X$, we have that for $x\in U(k)$, the specialization $A(x)$ is an $N$-by-$N$ matrix with entries in $k$. We let $f:X\times \mathbb {A}^N\dashrightarrow X\times \mathbb {A}^N$ be the rational endomorphism given by $(x,y)\mapsto (g(x), A(x)y)$. We prove that if the determinant of $A$ is nonzero and if there exists $x\in X(k)$ such that its orbit $\mathcal {O}_g(x)$ is Zariski dense in $X$, then either there exists a point $z\in (X\times \mathbb {A}^N)(k)$ such that its orbit $\mathcal {O}_f(z)$ is Zariski dense in $X\times \mathbb {A}^N$ or there exists a nonconstant rational function $\psi \in k(X\times \mathbb {A}^N)$ such that $\psi \circ f=\psi$. Our result provides additional evidence to a conjecture of Medvedev and Scanlon.References
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Additional Information
- Dragos Ghioca
- Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
- MR Author ID: 776484
- Email: dghioca@math.ubc.ca
- Junyi Xie
- Affiliation: IRMAR, Campus de Beaulieu, bâtiments 22-23 263 avenue du Général Leclerc, CS 74205 35042 Rennes Cédex France
- MR Author ID: 1079811
- Email: junyi.xie@univ-rennes1.fr
- Received by editor(s): June 29, 2017
- Received by editor(s) in revised form: January 17, 2018
- Published electronically: June 29, 2018
- Additional Notes: The first author was partially supported by a Discovery Grant from the National Sciences and Engineering Research Council of Canada, while the second author was partially supported by project “Fatou” ANR-17-CE40-0002-01.
- Communicated by: Matthew A. Papanikolas
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 4369-4387
- MSC (2010): Primary 37P15; Secondary 37P05
- DOI: https://doi.org/10.1090/proc/14104
- MathSciNet review: 3834665