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A conjecture strengthening the Zariski dense orbit problem for birational maps of dynamical degree one

Part of: Arithmetic algebraic geometry Abelian varieties and schemes Rings and algebras arising under various constructions Arithmetic and non-Archimedean dynamical systems

Published online by Cambridge University Press:  07 July 2022

Jason Bell Open the ORCID record for Jason Bell [Opens in a new window]  and
Dragos Ghioca
Jason Bell
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada e-mail: jpbell@uwaterloo.ca
Dragos Ghioca*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
*
e-mail: dghioca@math.ubc.ca
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Abstract

We formulate a strengthening of the Zariski dense orbit conjecture for birational maps of dynamical degree one. So, given a quasiprojective variety X defined over an algebraically closed field K of characteristic $0$ , endowed with a birational self-map $\phi $ of dynamical degree $1$ , we expect that either there exists a nonconstant rational function $f:X\dashrightarrow \mathbb {P} ^1$ such that $f\circ \phi =f$ , or there exists a proper subvariety $Y\subset X$ with the property that, for any invariant proper subvariety $Z\subset X$ , we have that $Z\subseteq Y$ . We prove our conjecture for automorphisms $\phi $ of dynamical degree $1$ of semiabelian varieties X. Moreover, we prove a related result for regular dominant self-maps $\phi $ of semiabelian varieties X: assuming that $\phi $ does not preserve a nonconstant rational function, we have that the dynamical degree of $\phi $ is larger than $1$ if and only if the union of all $\phi $ -invariant proper subvarieties of X is Zariski dense. We give applications of our results to representation-theoretic questions about twisted homogeneous coordinate rings associated with abelian varieties.

Keywords

Semiabelian varietiesendomorphismsdynamical degreedense orbitsDixmier–Moeglin equivalence

MSC classification

Primary: 11G10: Abelian varieties of dimension $> 1$ 14K12: Subvarieties 37P55: Arithmetic dynamics on general algebraic varieties
Secondary: 16S38: Rings arising from non-commutative algebraic geometry
Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 2 , June 2023 , pp. 477 - 491
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

The authors were partially supported by Discovery Grants from the National Science and Engineering Research Council of Canada.

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