Dimensão de Hausdorff para projeções de conjuntos de Cantor complexos dinamicamente definidos

Speaker: Alex Zamudio, UFRJ.

Date: 05 apr 2019, 16h15.

Place: Room 407, Bloco H, Campus Gragoatá, UFF.

Abstract: A classical theorem of Marstrand states that for any Borel subset $F \in R^2$

$$HD(πλ(F)) = min{1,HD(F)}$$

for almost all projections $πλ(x,y) = x + λy$ (with respect to Lebesgue measure in λ). Moreira was able to improve this theorem in the particular context of dynamically defined Cantor sets. He proved that given two dynamically defined Cantor sets $K₁, K₂ \in R^2$ satisfying some generic hypothesis, one has $HD(K₁+λK₂) = min {1, HD(K₁)+HD(K₂)}$ for all $λ ≠ 0$. We will talk about how Moreira's ideas can be generalized to Cantor sets in the complex plane, we will have a similar formula which holds for dynamically defined complex Cantor sets. In particular, this Cantor sets include Julia sets associated to quadratic maps $Q_c(z) = z² + c$ when the parameter c is not in the Mandelbrot set.