Height boundedness of (pre)periodic points of dominant rational self-maps

Periodic points of dominant rational self-maps on algebraic varieties have bounded height in several dynamically interesting cases, such as polarized endomorphisms or Henon maps. It is conjectured that the set of isolated periodic points of an automorphism of degree at least two on an affine space is a set of bounded height. We give a counterexample to this conjecture. 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. In the first part of the talk, I will explain the statements (maybe a little bit of the ideas behind the constructions as well) and also introduce some good properties of cohomologically hyperbolic maps. In the second half of the talk, I will sketch the proof and discuss some open problems. This talk is based on a joint work with Kaoru Sano.