A rational map with infinitely many points of distinct arithmetic degrees

Let $f\text{\hspace{0.17em}⁣}:X⇢X$ be a dominant rational self-map of a smooth projective variety defined over $\bar{\mathbb{Q}}$. For each point $P\in X(\bar{\mathbb{Q}})$ 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 $X={\mathbb{P}}^4$.