On invariant rational functions under rational transformations

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