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:XP ^1 such that f° φ =f, or there exists a proper subvariety YX with the property that, for any invariant proper subvariety ZX, we have that ZY . 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.