Periodic points and arithmetic degrees of certain rational self-maps

TL;DR

This paper investigates the arithmetic properties of periodic points and dynamical degrees of certain rational self-maps, providing boundedness results, verifying conjectures in new cases, and establishing the existence of dense orbits.

Contribution

It offers new boundedness results for heights of periodic points and verifies the Kawaguchi--Silverman conjecture for specific rational self-maps, linking it to the dynamical Mordell--Lang conjecture.

Findings

Boundedness of heights of periodic points for certain rational self-maps.

Verification of the Kawaguchi--Silverman conjecture in new cases.

Existence of Zariski dense orbits in these dynamical systems.

Abstract

Consider a cohomologically hyperbolic birational self-map defined over the algebraic numbers, for example, a birational self-map in dimension two with the first dynamical degree greater than one, or in dimension three with the first and the second dynamical degrees distinct. We give a boundedness result about heights of its periodic points. This is motivated by a conjecture of Silverman for polynomial automorphisms of affine spaces. We also study the Kawaguchi--Silverman conjecture concerning dynamical and arithmetic degrees for certain rational self-maps in dimension two. In particular, we reduce the problem to the dynamical Mordell--Lang conjecture and verify the Kawaguchi--Silverman conjecture for some new cases. As a byproduct of the argument, we show the existence of Zariski dense orbits in these cases.