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

Bell, Jason  ; Ghioca, Dragos

Canadian mathematical bulletin, Tome 66 (2023) no. 2, pp. 477-491

Résumé

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.

Détail

DOI : 10.4153/S0008439522000479

Mots-clés : Semiabelian varieties, endomorphisms, dynamical degree, dense orbits, Dixmier–Moeglin equivalence