Arithmetic degree and its application to Zariski dense orbit conjecture

TL;DR

This paper establishes a connection between arithmetic degrees and dynamical degrees for rational self-maps on varieties, and applies this to prove cases of the Zariski dense orbit conjecture, especially for threefolds with certain dynamical degree conditions.

Contribution

It introduces a method to relate arithmetic degrees to dynamical degrees and proves the Zariski dense orbit conjecture for specific birational maps over algebraic numbers.

Findings

Existence of points with arithmetic degree close to the first dynamical degree.

Zariski dense orbit conjecture holds when the first dynamical degree exceeds the third.

Confirmed the conjecture for threefolds with first dynamical degree greater than one.

Abstract

We prove that for a dominant rational self-map $f$ on a quasi-projective variety defined over $\overline{\mathbb{Q}}$, there is a point whose $f$-orbit is well-defined and its arithmetic degree is arbitrarily close to the first dynamical degree of $f$. As an application, we prove that Zariski dense orbit conjecture holds for a birational map defined over $\overline{\mathbb{Q}}$ whose first dynamical degree is strictly larger than its third dynamical degree. In particular, the conjecture holds for birational maps on threefolds whose first dynamical is degree greater than $1$.