Dynamical Degree and Arithmetic Degree of Endomorphisms on Product Varieties

Abstract

For a dominant rational self-map on a smooth projective variety defined over a number field, Shu Kawaguchi and Joseph H. Silverman conjectured that the dynamical degree is equal to the arithmetic degree at a rational point whose forward orbit is well-defined and Zariski dense. We give some examples of self-maps on product varieties and rational points on them for which the Kawaguchi-Silverman conjecture holds.

12 pages

References in corpus (2)

• The dynamical Mordell-Lang problem for etale maps
• On upper bounds of arithmetic degrees

Cited by in corpus (12)

• The Geometric Dynamical Northcott and Bogomolov Properties
• Non-density of points of small arithmetic degrees
• The existence of Zariski dense orbits for endomorphisms of projective surfaces (with an appendix in collaboration with Thomas Tucker)
• Kawaguchi-Silverman conjecture for endomorphisms on rationally connected varieties admitting an int-amplified endomorphism
• Arithmetic and dynamical degrees of self-morphisms of semi-abelian varieties
• A $1$-cohomologically hyperbolic birational map of ${\mathbb{P}}^3$, with a transcendental arithmetic degree
• Canonical heights for abelian group actions of maximal dynamical rank
• Advances in the equivariant minimal model program and their applications in complex and arithmetic dynamics
• Kawaguchi-Silverman conjecture for endomorphisms on several classes of varieties
• Arithmetic degrees for dynamical systems over function fields of characteristic zero
• Effective Eigendivisors and the Kawaguchi-Silverman Conjecture
• Zariski density of points with maximal arithmetic degree for surfaces