A rational map with infinitely many points of distinct arithmetic degrees
Abstract
Let $f \colon X \dashrightarrow X$ be a dominant rational selfmap of a smooth projective variety defined over $\overline{\mathbb Q}$. For each point $P\in X(\overline{\mathbb Q})$ whose forward $f$orbit is welldefined, 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 welldefined and that, as $P$ varies, the set of values obtained by $\alpha_f(P)$ is finite. Based on constructions of BedfordKim and McMullen, we give a counterexample to this conjecture when $X=\mathbb P^4$.
 Publication:

arXiv eprints
 Pub Date:
 August 2018
 arXiv:
 arXiv:1809.00047
 Bibcode:
 2018arXiv180900047L
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Number Theory
 EPrint:
 5 pages