A conjecture strengthening the Zariski dense orbit problem for birational maps of dynamical degree one. Issue 2 (7th June 2023) - Bell, Jason; Ghioca, Dragos
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 $φ $ of dynamical degree $1$, we expect that either there exists a nonconstant rational function $f:X\dashrightarrow \mathbb {P} ^1$ such that $f° φ =f$, 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 $φ $ of dynamical degree $1$ of semiabelian varieties X . Moreover, we prove a related result for regular dominant self-maps $φ $ of semiabelian varieties X : assuming that $φ $ does not preserve a nonconstant rational function, we have that the dynamical degree of $φ $ is larger than $1$ if and only if the union of all $φ $ -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.