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

Yohsuke Matsuzawa

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arXiv · 2407.03097

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

Yohsuke Matsuzawa

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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.

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Yohsuke Matsuzawa. 2024-07-03. Existence of arithmetic degrees for generic orbits and dynamical Lang-Siegel problem. https://arxiv.org/abs/2407.03097

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