東京大学大学院数理科学研究科

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数理談話会

日時: 2009年7月17日(金) 16:30〜17:30
会場: 東京大学 大学院数理科学研究科 002号室

講演者

Nessim Sibony 氏 (University of Paris-Sud)


講演題目

Holomorphic dynamics in several variables: equidistribution problems and statistical properties



講演概要

The main problem in the dynamical study of a map is to understand the long term behavior of orbits. The abstract theory of non uniformly hyperbolic systems is well understood but it is very difficult to decide when a given system is non uniformly hyperbolic and to study it's sharp ergodic properties. Holomorphic dynamics in several variables provide large classes of examples of non uniformly hyperbolic systems. One can compute the entropy, construct a measure of maximal entropy and study the sharp statistical properties: central limit theorem, large deviations and exponential decay of correlations. It is also possible to prove sharp equidistribution results for preimages of analytic sets of arbitrary dimension. The main tools are: pluripotential theory, analytic geometry, and good estimates from PDE. These systems appear naturally if we apply Newton's method to localise the common zeros of of polynomial equations in several variables. In the study of polynomial automorphisms of complex Euclidean spaces, or automorphisms of compact K\"ahler manifolds.