We determine when the equidistribution property for possibly moving targets holds for a rational function of degree more than one on the projective line over an algebraically closed field of any characteristic and complete with respect to a non-trivial absolute value. This characterization could be useful in the positive characteristic case. Based on a variational argument, we give a purely local proof of the adelic equidistribution theorem for possibly moving targets, which is due to Favre and...

In this article we prove an analogue of the equidistribution of preimages theorem from complex dynamics for maps of good reduction over nonarchimedean fields. While in general our result is only a partial analogue of the complex equidistribution theorem, for most maps of good reduction it is a complete analogue. In the particular case when the nonarchimedean field in question is equipped with the trivial absolute value, we are able to supply a strengthening of the theorem, namely that the preimages...

Given a rational function $\varphi \left(T\right)$ on ${\mathbb{P}}^1$ of degree at least 2 with coefficients in a number field $k$, we show that for each place $v$ of $k$, there is a unique probability measure ${\mu }_{\varphi ,v}$ on the Berkovich space ${\mathbb{P}}_{\mathrm{Berk},\mathrm{v}}^1/{ℂ}_v$ such that if $\left\{z_n\right\}$ is a sequence of points in ${\mathbb{P}}^1\left(\overline{k}\right)$ whose $\varphi$-canonical heights tend to zero, then the $z_n$’s and their $\mathrm{Gal}(\overline{k}/k)$-conjugates are equidistributed with respect to ${\mu }_{\varphi ,v}$.The proof uses a polynomial lift $F(x,y)=(F_1(x,y),F_2(x,y))$ of $\varphi$ to construct a two-variable Arakelov-Green’s function $g_{\varphi ,v}(x,y)$ for each $v$. The measure ${\mu }_{\varphi ,v}$ is obtained by taking the...

Let K be a complete, algebraically closed nonarchimedean valued field, and let φ(z) ∈ K(z) have degree d ≥ 2. We study how the resultant of φ varies under changes of coordinates. For γ ∈ GL₂(K), we show that the map $\gamma \mapsto ord\left(Res\left({\phi }^{\gamma }\right)\right)$ factors through a function $ordRe{s}_{\phi }\left(·\right)$ on the Berkovich projective line, which is piecewise affine and convex up. The minimal resultant is achieved either at a single point in $P{¹}_K$, or on a segment, and the minimal resultant locus is contained in the tree in $P{¹}_K$ spanned by the fixed points and poles...