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

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.-> P-1 such that f (degrees) phi = f, or there exists a proper subvariety Y subset of X with the property that, for any invariant proper subvariety Z subset of X, we have that Z subset of Y. We proveour conjecture for automorphisms phi of dynamical degree 1 of semiabelian varietiesX. Moreover, we prove a related result for regular dominant self-maps phi of semiabelian varieties X: assuming that. 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.