A conjecture strengthening the Zariski dense orbit problem for birational maps of dynamical degree one. Issue 2 (7th June 2023)

• Record Type: Journal Article
• Authors: Bell, Jason; Ghioca, Dragos
• Is Part Of: Canadian mathematical bulletin =. Volume 66:Issue 2(2023)
• Page Start: 477
• Page End: 491
• Publication Date: 2023-06-07
• Subjects: 11G10 -- 14K12 -- 37P55 -- 16S38; Semiabelian varieties -- endomorphisms -- dynamical degree -- dense orbits -- Dixmier–Moeglin equivalence; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/S0008439522000479
• Languages: English
• ISSNs: 0008-4395

Abstract

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.