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Equidistribution of small points, rational dynamics, and potential theory. (English) Zbl 1234.11082
Summary: Given a rational function $$\varphi (T)$$ 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 $$\mathbb{P}^1_{\text{Berk},v} / \mathbb{C}_v$$ such that if $$\{ z_n \}$$ is a sequence of points in $$\mathbb{P}^1(\overline{k})$$ whose $$\varphi$$-canonical heights tend to zero, then the $$z_n$$’s and their $$\text{Gal}(\overline{k}/k)$$-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 Berkovich space Laplacian of $$g_{\varphi ,v}(x,y)$$. The main ingredients in the proof are an energy minimization principle for $$g_{\varphi ,v}(x,y)$$ and a formula for the homogeneous transfinite diameter of the $$v$$-adic filled Julia set $$K_{F,v} \subset \mathbb{C}_v^2$$ for each place $$v$$.

##### MSC:
 11G50 Heights 37P50 Dynamical systems on Berkovich spaces 37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010) 31C15 Potentials and capacities on other spaces
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