  A binding group theorem is proved in the context of quantifier-free
internality to the fixed field in difference-closed fields of characteristic
zero. This is articulated as a statement about the birational geometry of
isotrivial algebraic dynamical systems, and more generally isotrivial
$\sigma$-varieties. It asserts that if $(V,\phi)$ is an isotrivial
$\sigma$-variety then a certain subgroup of the group of birational
transformations of $V$, namely those that preserve all the relations between
$(V,\phi)$ and the trivial dynamics on the affine line, is in fact an algebraic
group. Several application are given including new special cases of the Zariski
Dense Orbit Conjecture and the Dixmier-Moeglin Equivalence Problem in algebraic
dynamics, as well as finiteness results about the existence of nonconstant
invariant rational functions on cartesian powers of $\sigma$-varieties. These
applications give algebraic-dynamical analogues of recent results in
differential-algebraic geometry.
