Seminários de Sistemas Dinâmicos.

Resumo: A classical theorem of Marstrand states that for any Borel subset F ?R2 HD(p?(F)) = min{1,HD(F)}, for almost all projection p?(x,y) = x+?y (with respect to Lebesgue measure in ?). Moreira was able to improve this theorem in the particular context of dynamically defined Cantor sets. He proved that given dynamically defined Cantor sets K1,K2 ? R satisfying some generic hypothesis one has HD(K1 +?·K2) =min{1,HD(K1)+HD(K2)}, for all ?= 0. We will talk about how Moreiras ideas can be generalized to Cantor sets in the complex plane, in particular we will have a similar formula which holds for dynamically defined complex Cantor sets. In particular, this Cantor sets include Julia sets associated to quadratic maps Qc(z) = z2 + c when the parameter c is not in the Mandelbrot set.