A rational dynamical system consists of a variety X and a rational (algebraic) map from X to itself. We apply model theory to study a special class of such dynamical systems (and more general "twisted" versions), the *isotrivial* ones, showing, for instance, that if the system has no suitably defined "first integrals", then it has only finitely many maximal proper invariant subvarieties (the Dixmier--Moeglin problem).

This result, as well as others, is a corollary of the main theorem, which states that the birational automorphism group of such a system is an algebraic group. This, in turn, is a special case of the "binding group" theorem in model theory. As the relevant theory, ACFA, fails to have quantifier elimination, we require a quantifier-free, type-definable variant of the classical results.

This is joint work with Rahim Moosa from the University of Waterloo.