Equidistribution of repelling fixed points for iterates of rational functions
Prove that for every rational function R over an algebraically closed, complete non-archimedean field, with degree at least two, the degree-weighted measures supported on the repelling fixed points of the iterates R^n, normalized by (deg R^n)^{-1}, converge weakly to the equilibrium measure of R as n tends to infinity.
References
Their Question 1 then asks whether equidistribution holds for repelling periodic points. In , the same authors rephrase this question in a more precise way in their Conjecture 1, which we state here below. Let $R\in K(z)$ be a rational function with $\deg R \geq 2$, denote by $(Rn)$ the set of repelling fixed points of $Rn$, and let $\sigma_R = \sum_{\zeta \in (R)} (\deg_R \zeta)\times\delta_\zeta$. Then $(\deg Rn){-1}\sigma_{Rn} \xrightarrow[n\to\infty]{} \nu_R$ weakly.