  Let $G$ be a semiabelian variety defined over an algebraically closed field
$K$ of characteristic $0$. Let $\Phi\colon G\dashrightarrow G$ be a dominant
rational self-map. Assume that an iterate $\Phi^m \colon G \to G$ is regular
for some $m \geqslant 1$ and that there exists no non-constant homomorphism
$\tau: G\to G_0$ of semiabelian varieties such that $\tau\circ \Phi^{m k}=\tau$
for some $k \geqslant 1$. We show that under these assumptions $\Phi$ itself
must be a regular. We also prove a variant of this assertion in prime
characteristic and present examples showing that our results are sharp.
