Invariant rational functions under rational transformations

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.