Growth of generalized greatest common divisors along orbits of self-rational maps on projective varieties
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913930021961728 |
|---|---|
| author | Matsuzawa, Yohsuke |
| author_facet | Matsuzawa, Yohsuke |
| contents | Consider a dominant rational self-map $f$ on a smooth projective variety $X$ defined over $\overline{\mathbb{Q}}$. We prove that \begin{align} \lim_{n \to \infty} \frac{h_{Y}(f^{n}(x))}{h_{H}(f^{n}(x)) } = 0, \end{align} where $h_{Y}$ is a height associated with a closed subscheme $Y \subset X$ of codimension $c$, $h_{H}$ is any ample height on $X$, and $x \in X(\overline{\mathbb{Q}})$ is a point with well-defined orbit, under the following assumptions: (1) either $f$ is a morphism, or $Y$ is pure dimensional, regularly embedded in $X$, and contained in the locus over which all iterates of $f$ are finite; (2) the orbit of $x$ is generic; (3) $d_{c}(f)^{1/c} < α_{f}(x)$, where $d_{c}(f)$ is the $c$-th dynamical degree of $f$ and $ α_{f}(x)$ is the arithmetic degree of $x$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_05027 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Growth of generalized greatest common divisors along orbits of self-rational maps on projective varieties Matsuzawa, Yohsuke Algebraic Geometry Dynamical Systems Number Theory 37P15, 37P55, 11G50 Consider a dominant rational self-map $f$ on a smooth projective variety $X$ defined over $\overline{\mathbb{Q}}$. We prove that \begin{align} \lim_{n \to \infty} \frac{h_{Y}(f^{n}(x))}{h_{H}(f^{n}(x)) } = 0, \end{align} where $h_{Y}$ is a height associated with a closed subscheme $Y \subset X$ of codimension $c$, $h_{H}$ is any ample height on $X$, and $x \in X(\overline{\mathbb{Q}})$ is a point with well-defined orbit, under the following assumptions: (1) either $f$ is a morphism, or $Y$ is pure dimensional, regularly embedded in $X$, and contained in the locus over which all iterates of $f$ are finite; (2) the orbit of $x$ is generic; (3) $d_{c}(f)^{1/c} < α_{f}(x)$, where $d_{c}(f)$ is the $c$-th dynamical degree of $f$ and $ α_{f}(x)$ is the arithmetic degree of $x$. |
| title | Growth of generalized greatest common divisors along orbits of self-rational maps on projective varieties |
| topic | Algebraic Geometry Dynamical Systems Number Theory 37P15, 37P55, 11G50 |
| url | https://arxiv.org/abs/2507.05027 |