  Let $X$ be an algebraic variety equipped with a dominant rational self-map
$\phi:X\to X$. A new quantity measuring the interaction of $(X,\phi)$ with
trivial dynamical systems is introduced; the stabilised algebraic dimension of
$(X,\phi)$ captures the maximum number of new algebraically independent
invariant rational functions on the cartesian product of $(X, \phi)$ and $(Y,
\psi)$, as $(Y,\psi)$ ranges over all algebraic dynamical systems. It is shown
that this birational invariant agrees with the maximum dimension of a dominant
equivariant rational image $(X',\phi')$ where $\phi'$ is part of an algebraic
group action on $X'$. As a consequence, it is deduced that if some cartesian
power of $(X,\phi)$ admits a nonconstant invariant rational function, then
already the second cartesian power does.
