Equidistribution of the crucial measures in non-Archimedean dynamics

TextLet K   be a complete, algebraically closed, non-Archimedean valued field, and let ϕ∈K(z)ϕ∈K(z) with deg⁡(ϕ)≥2deg⁡(ϕ)≥2. In this paper we consider the family of functions ordResϕn(x), which measure the resultant of ϕnϕn at points x   in PK1, the Berkovich projective line, and show that they converge locally uniformly to the diagonal values of the Arakelov–Green's function gμϕ(x,x)gμϕ(x,x) attached to the canonical measure of ϕ  . Following this, we are able to prove an equidistribution result for Rumely's crucial measures νϕnνϕn, each of which is a probability measure supported at finitely many points whose weights are determined by dynamical properties of ϕ.VideoFor a video summary of this paper, please visit https://youtu.be/YCCZD1iwe00.