Periodic points and arithmetic degrees of certain rational self-maps

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 affine automorphisms. We also study the Kawaguchi-Silverman conjecture concerning the dynamical and the arithmetic degrees for birational 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.