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We formulate a strengthening of the Zariski dense orbit conjecture forbirational 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 expectthat either there exists a non-constant 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 forautomorphisms $\phi$ of dynamical degree $1$ of semiabelian varieties $X$.Also, we prove a related result for regular dominant self-maps $\phi$ ofsemiabelian varieties $X$: assuming $\phi$ does not preserve a non-constantrational 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 representationtheoretic questions about twisted homogeneous coordinate rings associated toabelian varieties.

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