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.

Let X be a normal projective variety of dimension n and G an abelian group of automorphisms such that all elements of $G\setminus \{\operatorname {id}\}$ are of positive entropy. Dinh and Sibony showed that G is actually free abelian of rank $\le n - 1$ . The maximal rank case has been well understood by De-Qi Zhang. We aim to characterize the pair $(X, G)$ such that $\operatorname {rank} G = n - 2$ .

We prove a conjecture by Kawaguchi–Silverman on arithmetic and dynamical degrees, for self-morphisms of semi-abelian varieties. Moreover, we determine the set of the arithmetic degrees of orbits and the (first) dynamical degrees of self-morphisms of semi-abelian varieties.