Finiteness Properties of Pseudo-Hyperbolic Varieties

Motivated by Lang-Vojta's conjecture, we show that the set of dominant rational selfmaps of an algebraic variety over a number field with only finitely many rational points in any given number field is finite by combining Amerik's theorem for dynamical systems of infinite order with properties of Prokhorov-Shramov's notion of quasiminimal models. We also prove a similar result in the geometric setting by using again not only Amerik's theorem and Prokhorov-Shramov's notion of quasi-minimal model but also Weil's regularization theorem for birational self-maps and properties of dynamical degrees. Furthermore, in the geometric setting, we obtain an analogue of KobayashiOchiai's finiteness result for varieties of general type and thereby generalize Noguchi's theorem (formerly Lang's conjecture).