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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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