On the height boundedness of periodic and preperiodic points of dominant rational self-maps on projective varieties

We give a counterexample to the following conjecture: the set of isolated periodic points of an automorphism of degree at least two on an affine space is a set of bounded height. As a positive result, we prove that any cohomologically hyperbolic dominant rational self-map on a projective variety admits a non-empty Zariski open subset on which the set of periodic points is height bounded. Concerning preperiodic points, we give an example suggesting that the same statement may fail.