Abstract:

In the first part of the talk, we will review some of the latest results which have contributed to understand the geometrical structure of metric measure spaces supporting Poincare inequalities. We focus our attention in a classical fractal: the Sierpinski carpet endowed with its associated Hausdorff measure.

In the second part, we will characterize the slopes of nontrivial line segments contained in self-similar Sierpinski carpets. The set of slopes is related to Farey sequences and the dynamics of punctured square toral billiards. As a consequence, we deduce conclusions about the collection of everywhere differentiable curves contained in such carpets.