**Abstract:** |
Let be a field and let be the separable closure of in the algebraic closure kc of . An algebraic zero divisor (P) on the classical projective line over kc, which is the zeros divisor on of a homogeneous polynomial of two variables over , generalizes both the Galois conjugacy class of a point in and the roots divisor of the equation on for distinct rational functions , satisfying min{deg , deg} > 0. The mass of (P)×(P) on the diagonal of x, called the diagonal of (P), play a non-trivial role in the studying such an algebraic zero divisor (P) more general than Galois conjugacy classes. If is a product formula field, i.e., a field equipped with adelic places satisfying the product formula, then we can also associate to the algebraic divisor (P) a height ((P)) with respect to an adelic family g of normalized continuous weights (or potentials) on the Berkovich projective lines for all places on , in a similar way to define the classical heights of Galois conjugacy classes. In this talk, we give a generalization of the adelic equidistribution theorem due to Baker-Rumely, Favre-Rivera-Letelier, and Chambert-Loir and its quantitative version due to Favre-Rivera-Leteier for a sequence of Galois conjugacy classes on having small heights. This generalization is suitable for studying both the dynamics and the value distribution of the iteration of a rational function of degree more than one since the roots divisor =a for a rational function and large enough is not necessarily a Galois conjugacy classes in but always an algebraic zeros divisor on for a fixed number field . |
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