On invariant rational functions under rational transformations

Let X be an algebraic variety equipped with a dominant rational map \(\phi :X\dashrightarrow X\). 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 \((X\times Y,\phi \times \psi )\), as \(\psi :Y\dashrightarrow Y\) ranges over all dominant rational maps on algebraic varieties. It is shown that this birational invariant agrees with the maximum \(\dim X'\) where \((X,\phi )\dashrightarrow (X',\phi ')\) is a dominant rational equivariant map and \(\phi '\) 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.