Equidistribution of Dynamically Small Subvarieties over the Function Field of a Curve

TL;DR

This paper proves that subvarieties with small height in algebraic dynamical systems over function fields are equidistributed in Berkovich spaces, with applications to the distribution of preperiodic points and small height points.

Contribution

It establishes an equidistribution theorem for small-height subvarieties over function fields, extending arithmetic dynamics to Berkovich analytic spaces.

Findings

Small-height subvarieties are equidistributed at each place.

Applications include non-Zariski density of preperiodic points.

Develops arithmetic intersection theory for this setting.

Abstract

For a projective variety X defined over a field K, there is a special class of self-morphisms of X called algebraic dynamical systems. In this paper we take K to be the function field of a smooth curve and prove that at each place of K, subvarieties of X of dynamically small height are equidistributed on the associated Berkovich analytic space. We carefully develop all of the arithmetic intersection theory needed to state and prove this theorem, and we present several applications on the non-Zariski density of preperiodic points and of points of small height in field extensions of bounded degree.