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

Yohsuke Matsuzawa

TL;DR

This work proves the existence of arithmetic degrees for points with generic orbits under dominant rational maps on projective varieties, and extends the result to étale morphisms on quasi-projective varieties. It shows that the arithmetic degree $\alpha_f(x)$ exists for generic orbits and equals a dynamical degree-derived value $\mu_\ell(f)$, linking height growth to dynamical degrees. The authors apply these ideas to the dynamical Lang-Siegel problem, demonstrating that local height contributions along orbits grow slowly when a zero-dimensional subscheme is considered, and they establish Banach-density results when local height growth is fast on a subset. The paper blends height theory, dynamical degrees, and Roth-type Diophantine arguments to derive new constraints on growth along orbits and to quantify exceptional subsets, with implications for Lang-Siegel-type phenomena in dynamical systems.

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.

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

TL;DR

This work proves the existence of arithmetic degrees for points with generic orbits under dominant rational maps on projective varieties, and extends the result to étale morphisms on quasi-projective varieties. It shows that the arithmetic degree exists for generic orbits and equals a dynamical degree-derived value , linking height growth to dynamical degrees. The authors apply these ideas to the dynamical Lang-Siegel problem, demonstrating that local height contributions along orbits grow slowly when a zero-dimensional subscheme is considered, and they establish Banach-density results when local height growth is fast on a subset. The paper blends height theory, dynamical degrees, and Roth-type Diophantine arguments to derive new constraints on growth along orbits and to quantify exceptional subsets, with implications for Lang-Siegel-type phenomena in dynamical systems.

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.
Paper Structure (9 sections, 25 theorems, 174 equations)

This paper contains 9 sections, 25 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Height growth and arithmetic degree
  3. Dynamical Lang-Siegel problem for rational map
  4. Banach density and dynamical Lang-Siegel problem
  5. Roth's theorem
  6. Backward limit of upper semicontinuous functions under rational maps
  7. Finite loci of rational map
  8. Backward limit of upper semicontinuous functions
  9. Multiplicities

Key Result

Theorem 1.3

When the author wrote the first version of this paper, Junyi Xie informed me that he was also aware of this theorem. The author appreciates his permission to include this theorem in this paper. Let $f \colon X \dashrightarrow X$ be a dominant rational map on a projective variety $X$ defined over $\o exists.

Theorems & Definitions (68)

  • Conjecture 1.1: Kawaguchi-Silverman conjecture Sil12KS16b
  • Definition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Remark 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Definition 2.1
  • ...and 58 more