Growth of local height functions along orbits of self-morphisms on projective varieties

TL;DR

This paper investigates the asymptotic behavior of local height functions along orbits of self-morphisms on projective varieties, establishing conditions under which certain limits vanish and deriving dynamical analogs of Lang-Siegel theorems.

Contribution

It provides new geometric criteria for the vanishing of height function limits and extends Silverman's classical results to higher dimensions, under certain conjectural assumptions.

Findings

Limit of local height sums is zero under specified conditions.

Proves dynamical Lang-Siegel type theorems for higher-dimensional varieties.

Unconditional results for zero-dimensional subschemes, conditional on Vojta's conjecture for higher dimensions.

Abstract

In this paper, we consider the limit $ \lim_{n \to \infty} \sum_{v\in S} \lambda_{Y,v}(f^{n}(x))/h_{H}(f^{n}(x)) $ where $f \colon X \longrightarrow X$ is a surjective self-morphism on a smooth projective variety $X$ over a number field, $S$ is a finite set of places, $ \lambda_{Y,v}$ is a local height function associated with a proper closed subscheme $Y \subset X$, and $h_{H}$ is an ample height function on $X$. We give a geometric condition which ensures that the limit is zero, unconditionally when $\dim Y=0$ and assuming Vojta's conjecture when $\dim Y\geq1$. In particular, we prove (one is unconditional, one is assuming Vojta's conjecture) Dynamical Lang-Siegel type theorems, that is, the relative sizes of coordinates of orbits on $\mathbb{P}^{N}$ are asymptotically the same with trivial exceptions. These results are higher dimensional generalization of Silverman's classical result.