Ergodic theory of rational fractions on ultrametric bodies
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Date
2010
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Publisher
WILEY
Abstract
We make the first steps towards an understanding of the ergodic properties of a rational map defined over a complete algebraically closed non-archimedean field. For such a rational map R, we construct a natural invariant probability measure (R) which represents the asymptotic distribution of preimages of non-exceptional points. We show that this measure is exponentially mixing, and satisfies the central limit theorem. We prove some general bounds on the metric entropy of (R), and on the topological entropy of R. We finally prove that rational maps with vanishing topological entropy have potential good reduction.
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Keywords
TRANSFER OPERATOR, PERIODIC POINTS, DYNAMICS, EQUIDISTRIBUTION, ITERATION, HEIGHTS