  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.
