Equidistribution in non-archimedean parameter curves towards the activity measures

Abstract

For every pair of an analytic family $f=f_{t}$ of endomorphisms of degree $>1$ of the Berkovich projective line $\mathbb{P}^{1,\mathrm{an}}$ over an algebraically closed and complete non-trivially valued field $K$ and an analytically marked point $a=a(t)$ in $\mathbb{P}^{1,\mathrm{an}}$ both parametrized by a domain $V$ in the Berkovich analytification of a smooth projective algebraic curve $C/K$, we establish the equidistribution of the averaged pullbacks of any value in $\mathbb{P}^{1,\mathrm{an}}$ but a subset of logarithmic capacity 0 under the sequence of the morphisms $a_{n}=a_{n}(t)=f_{t}^{n}(a(t)):V\to\mathbb{P}^{1,\mathrm{an}}$, towards the activity measure $\mu_{(f,a)}$ on $V$ associated with $f$ and $a$.