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Entropy of quadratic polynomials and dimension of sections of the Mandelbrot set

Le : 18/12/2012 16h15
Par : Guilio Tiozzo (Universite de Harvard)
Lieu : I103
Lien web :
Résumé : The topological entropy of the action of a quadratic polynomial f_c(z) = z^2 + c on the Riemann sphere is constant, independent of the parameter c. However, if one restricts the action to the corresponding Hubbard tree, then the entropy varies very interestingly with the parameter, and its variation reflects the geometry of the Mandelbrot set. We prove, for real parameters, a formula relating the topological entropy to the Hausdorff dimension of the set of external rays landing on the real slice of the Mandelbrot set, to the right of c. The result can be generalized to some non-real veins, by looking at the entropy of Hubbard trees.