Zariski dense orbits for regular self-maps of tori in positive characteristic
type: Item
instance of: scholarly article
title: Zariski dense orbits for regular self-maps of tori in positive characteristic
zbMATH Open document ID: 1490.14077
author: Dragos Ghioca
author: Sina Saleh
published in: The New York Journal of Mathematics
publication date: 2022-02-16T00:00:00Z
full work available at URL: 27-49.html
review text: The paper under review formulates a characteristic \(p\) analogue of a dynamical conjecture of Zhang, Medvedev-Scanlon, and Amerik-Campana, stating that for any dominant rational self-map of a quasiprojective variety, either there is some well defined dense orbit, or else there is some invariant non-constant rational function. The naive analogue of this conjecture fails in characteristic \(p\) for the Frobenius endomorphism. Thus, the authors introduce a third possibility, which is that some power of the endomorphism might be conjugate to the Frobenius endomorphism on some (possibly different) quasiprojective variety. The authors note that their conjecture is essentially known when the base field is uncountable, and prove it themselves in the case that the quasiprojective variety is \(\mathbb{G}_m^N\).
reviewed by: David McKinnon
Mathematics Subject Classification ID: 14K15
Mathematics Subject Classification ID: 14G05
zbMATH DE Number: 7474324
zbMATH Keywords: Zariski dense orbits
zbMATH Keywords: Medvedev-Scanlon conjecture
zbMATH Keywords: Mordell-Lang theorem in positive characteristic for tori
MaRDI profile type: Publication
cites work: Meromorphic fibrations on certain manifolds with trivial canonical bundle
cites work: On a dynamical version of a theorem of Rosenlicht
cites work: On the Medvedev-Scanlon conjecture for minimal threefolds of nonnegative Kodaira dimension
cites work: The dynamical Mordell-Lang conjecture
cites work: THE DYNAMICAL MORDELL–LANG CONJECTURE FOR ENDOMORPHISMS OF SEMIABELIAN VARIETIES DEFINED OVER FIELDS OF POSITIVE CHARACTERISTIC
cites work: Q4322758
cites work: The isotrivial case in the Mordell-Lang Theorem
cites work: Density of orbits of endomorphisms of commutative linear algebraic groups
cites work: A SPARSITY RESULT FOR THE DYNAMICAL MORDELL–LANG CONJECTURE IN POSITIVE CHARACTERISTIC
cites work: Density of orbits of dominant regular self-maps of semiabelian varieties
cites work: Density of orbits of endomorphisms of abelian varieties
cites work: Algebraic dynamics of skew-linear self-maps
cites work: The Mordell-Lang conjecture for function fields
cites work: Q4109766
cites work: Exponential diophantine equations
cites work: Invariant varieties for polynomial dynamical systems
cites work: Q5700780
cites work: F -structures and integral points on semiabelian varieties over finite fields
cites work: Q4424890
cites work: Integral Points on Subvarieties of Semiabelian Varieties, II
cites work: Q5434295
Recommended article: Zariski dense orbits for regular self-maps of split semiabelian varieties in positive characteristic
Recommended article: Zariski dense orbits for regular self-maps on split semiabelian varieties
Recommended article: Density of orbits of dominant regular self-maps of semiabelian varieties
Recommended article: The existence of Zariski dense orbits for polynomial endomorphisms of the affine plane
Recommended article: A conjecture strengthening the Zariski dense orbit problem for birational maps of dynamical degree one
Recommended article: Zariski dense orbits for endomorphisms of a power of the additive group scheme defined over finite fields
Recommended article: The Zariski-Lipman conjecture for toric varieties
Recommended article: Affine Endomorphisms with a Dense Orbit
Recommended article: Q4369157
Recommended article: Q4696928
description: scientific article; zbMATH DE number 7474324