On invariant rational functions under rational transformations (Q6555696)

The authors consider rational dynamical systems \((X,\phi)\) over an algebraically closed field \(k\), where \(X\) is an irreducible quasi-projective variety over \(k\) and \(\phi:X\dashrightarrow X\) is a dominant rational map over \(k\). The main result of the paper is that two dimensions defined by the authors coincide. The first one is 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 called stabilised algebraic dimension of \((X, \phi)\) and denoted by \(\mathrm{sadim}(X, \phi)\). The second dimension is the maximum \(n\geq 0\) such that there is a translational dynamical system \((Y,\psi)\) over \(k\) of dimension \(n\), and a dominant rational equivariant map \((X, \phi) \dashrightarrow (Y,\psi)\). It is called the translation dimension of \((X, \phi)\) and denoted by tdim\((X, \phi)\).