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

2024/07/03 by Yohsuke Matsuzawa, Matsuzawa, Yohsuke · 2 citations
Computer Science · Mathematics · #37P15 #37P55 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Coding theory and cryptography #Dynamical Systems (math.DS) #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2407.03097

openalex publication_date 2024/07/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 étale 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.

Cited by

• Arithmetic Degrees are Cohomological Lyapunov Multipliers
• Growth of generalized greatest common divisors along orbits of self-rational maps on projective varieties

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