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

TL;DR

This paper disproves a conjecture about bounded heights of isolated periodic points for certain automorphisms, but shows that for cohomologically hyperbolic maps on projective varieties, periodic points are height bounded on some open subset.

Contribution

It provides a counterexample to a conjecture and establishes a positive result for cohomologically hyperbolic maps on projective varieties.

Findings

Counterexample to the bounded height conjecture for affine automorphisms.

Proof that cohomologically hyperbolic maps have height-bounded periodic points on some open subset.

Example indicating potential failure of boundedness for preperiodic points.

Abstract

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.