Dynamical Degree and Arithmetic Degree of Endomorphisms on Product Varieties

TL;DR

This paper investigates the Kawaguchi-Silverman conjecture relating dynamical and arithmetic degrees for self-maps on product varieties, providing examples where the conjecture holds, thus advancing understanding in arithmetic dynamics.

Contribution

The paper offers new examples of self-maps on product varieties that satisfy the Kawaguchi-Silverman conjecture, supporting its validity in broader contexts.

Findings

Examples confirming the conjecture on specific product varieties.

Evidence that the dynamical degree equals the arithmetic degree in these cases.

Enhanced understanding of the conjecture's applicability to complex 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.