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

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 $f:X⟶X$ 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 $\dim Y\ge 1$. 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.

arXiv comment28 pages