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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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The existence of Zariski dense orbits for endomorphisms of projective surfaces
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by Junyi Xie; with an appendix in collaboration with Thomas Tucker HTML | PDF
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Let $f$ be a dominant endomorphism of a smooth projective surface $X$ over an algebraically closed field $\mathbf {k}$ of characteristic $0$. We prove that if there is no rational function $H \in \mathbf {k}(X)$ such that $H \circ f = H$, then there exists a point $x \in X(\mathbf {k})$ such that the forward orbit of $x$ under $f$ is Zariski dense in $X$. This result gives us a positive answer to the Zariski dense orbit conjecture for endomorphisms of smooth projective surfaces.

We also define a new topology on varieties over algebraically closed fields with finite transcendence degree over $\mathbb {Q}$, which we call “the adelic topology”. This topology is stronger than the Zariski topology, and an irreducible variety is still irreducible in this topology. Using the adelic topology, we propose an adelic version of the Zariski dense orbit conjecture that is stronger than the original one and quantifies how many such orbits there are. We also prove this adelic version for endomorphisms of smooth projective surfaces, for endomorphisms of abelian varieties, and for split polynomial maps. This yields new proofs of the original conjecture in the latter two cases.

In Appendix A, we study endomorphisms of $k$-affinoid spaces. We show that for certain endomorphisms $f$ on a $k$-affinoid space $X$, the attractor $Y$ of $f$ is a Zariski closed subset and that the dynamics of $f$ is semi-conjugate to its restriction to $Y$. A special case of this result is used in the proof of the main theorem.

In Appendix B, written in collaboration with Thomas Tucker, we prove the Zariski dense orbit conjecture for endomorphisms of $(\mathbb {P}^1)^N$.

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Additional Information
  • Junyi Xie
  • Affiliation: Beijing International Center for Mathematical Research, Peking University, Beijing 100871, China and Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France
  • MR Author ID: 1079811
  • ORCID: 0000-0001-7386-8810
  • Email:
  • Thomas Tucker
  • MR Author ID: 310767
  • ORCID: 0000-0002-8582-2198
  • Received by editor(s): November 29, 2019
  • Received by editor(s) in revised form: April 26, 2020, April 3, 2021, October 14, 2021, and January 8, 2022
  • Published electronically: April 18, 2022
  • Additional Notes: The author was partially supported by project “Fatou” ANR-17-CE40-0002-01 and PEPS CNRS
  • © Copyright 2022 American Mathematical Society
  • Journal: J. Amer. Math. Soc.
  • MSC (2020): Primary 37P55; Secondary 37P20
  • DOI: