  Motivated by Lang-Vojta's conjecture, we show that the set of dominant
rational self-maps 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 quasi-minimal models. We also prove a similar
result in the geometric setting by using again 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
Kobayashi-Ochiai's finiteness result for varieties of general type, and thereby
generalize Noguchi's theorem (formerly Lang's conjecture).
