  The Mordell-Lang conjecture describes the intersection of a finitely
generated subgroup with a closed subvariety of a semiabelian variety.
Equivalently, this conjecture describes the intersection of closed subvarieties
with the set of images of the origin under a finitely generated semigroup of
translations. We study the analogous question in which the translations are
replaced by algebraic group endomorphisms (and the origin is replaced by
another point). We show that the conclusion of the Mordell-Lang conjecture
remains true in this setting if either (1) the semiabelian variety is simple,
(2) the semiabelian variety is $A^2$, where $A$ is a one-dimensional
semiabelian variety, (3) the subvariety is a connected one-dimensional
algebraic subgroup, or (4) each endomorphism has diagonalizable Jacobian at the
origin. We also give examples showing that the conclusion fails if we make
slight modifications to any of these hypotheses.
