A rational map with infinitely many points of distinct arithmetic degrees

TL;DR

This paper constructs a specific example of a rational map on a projective variety that demonstrates the set of arithmetic degrees can be infinite, countering previous conjectures about their finiteness.

Contribution

It provides the first known counterexample to the Kawaguchi-Silverman conjecture on the finiteness of arithmetic degrees for rational maps.

Findings

Counterexample on $ ext{P}^4$ showing infinitely many arithmetic degrees

Disproves the conjecture that the set of arithmetic degrees is finite

Uses constructions from Bedford--Kim and McMullen

Abstract

Let $f \colon X \dashrightarrow X$ be a dominant rational self-map of a smooth projective variety defined over $\overline{\mathbb Q}$. For each point $P\in X(\overline{\mathbb Q})$ whose forward $f$-orbit is well-defined, Silverman introduced the arithmetic degree $\alpha_f(P)$, which measures the growth rate of the heights of the points $f^n(P)$. Kawaguchi and Silverman conjectured that $\alpha_f(P)$ is well-defined and that, as $P$ varies, the set of values obtained by $\alpha_f(P)$ is finite. Based on constructions of Bedford--Kim and McMullen, we give a counterexample to this conjecture when $X=\mathbb P^4$.