Existence of arithmetic degrees for generic orbits and dynamical Lang-Siegel problem

TL;DR

This paper establishes the existence of arithmetic degrees for generic orbits of dominant rational maps and étale morphisms, and applies these results to the dynamical Lang-Siegel problem, analyzing growth rates of local height functions.

Contribution

It proves the existence of arithmetic degrees for generic orbits and applies this to analyze growth of local height functions in dynamical systems.

Findings

Arithmetic degrees exist for generic orbits of dominant rational maps.

Local height functions grow slowly along orbits under certain conditions.

Subsets with fast-growing local height functions have Banach density zero.

Abstract

We prove the existence of the arithmetic degree for dominant rational self-maps at any point whose orbit is generic. As a corollary, we prove the same existence for \'etale morphisms on quasi-projective varieties and any points on it. We apply the proof of this fact to dynamical Lang-Siegel problem. Namely, we prove that local height function associated with zero-dimensional subscheme grows slowly along orbits of a rational map under reasonable assumption. Also if local height function associated with any proper closed subscheme grows fast on a subset of an orbit of a self-morphism, we prove that such subset has Banach density zero under some assumptions.