The paper under review formulates a characteristic \(p\) analogue of a dynamical conjecture of Zhang, Medvedev-Scanlon, and Amerik-Campana, stating that for any dominant rational self-map of a quasiprojective variety, either there is some well defined dense orbit, or else there is some invariant non-constant rational function.
The naive analogue of this conjecture fails in characteristic \(p\) for the Frobenius endomorphism. Thus, the authors introduce a third possibility, which is that some power of the endomorphism might be conjugate to the Frobenius endomorphism on some (possibly different) quasiprojective variety.
The authors note that their conjecture is essentially known when the base field is uncountable, and prove it themselves in the case that the quasiprojective variety is \(\mathbb{G}_m^N\).