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The existence of Zariski dense orbits for polynomial endomorphisms of the affine plane

Part of: Arithmetic and non-Archimedean dynamical systems Holomorphic mappings and correspondences

Published online by Cambridge University Press:  31 May 2017

Junyi Xie
Junyi Xie*
Affiliation:
Université de Rennes I, Campus de Beaulieu, bâtiment 22–23, 35042 Rennes cedex, France email junyi.xie@univ-rennes1.fr
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Abstract

In this paper we prove the following theorem. Let $f$ be a dominant polynomial endomorphism of the affine plane defined over an algebraically closed field of characteristic $0$. If there is no nonconstant invariant rational function under $f$, then there exists a closed point in the plane whose orbit under $f$ is Zariski dense. This result gives us a positive answer to a conjecture proposed by Medvedev and Scanlon, by Amerik, Bogomolov and Rovinsky, and by Zhang for polynomial endomorphisms of the affine plane.

Keywords

polynomial endomorphismorbitvaluative tree

MSC classification

Primary: 37P55: Arithmetic dynamics on general algebraic varieties 32H50: Iteration problems
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 8 , August 2017 , pp. 1658 - 1672
Copyright
© The Author 2017 

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