Arithmetic degrees and dynamical degrees of endomorphisms on surfaces

TL;DR

This paper proves the Kawaguchi-Silverman conjecture for surjective endomorphisms on smooth projective surfaces, establishing a link between dynamical and arithmetic degrees for rational points.

Contribution

It confirms the conjecture for surfaces and shows the existence of rational points with arithmetic degrees equal to the dynamical degree on higher-dimensional varieties.

Findings

Proves the Kawaguchi-Silverman conjecture for surfaces.

Shows existence of rational points with matching arithmetic and dynamical degrees.

Establishes Zariski dense sets of rational points with disjoint orbits for automorphisms.

Abstract

For a dominant rational self-map on a smooth projective variety defined over a number field, Kawaguchi and Silverman conjectured that the (first) dynamical degree is equal to the arithmetic degree at a rational point whose forward orbit is well-defined and Zariski dense. We prove this conjecture for surjective endomorphisms on smooth projective surfaces. For surjective endomorphisms on any smooth projective varieties, we show the existence of rational points whose arithmetic degrees are equal to the dynamical degree. Moreover, we prove that there exists a Zariski dense set of rational points having disjoint orbits if the endomorphism is an automorphism.