One-Dimensional and Multi-Dimensional Continued Fractions: Geometry & Topology

I will present geometric and topological properties of the Gauss map acting on irrational numbers with restricted continued expansion entries. I will also discuss geometric and topological properties of multidimensional Gauss iterated function systems. Geometrically, I will describe in detail their Hausdorff and packing measures, along with the structure of the Hausdorff dimensions of their subsystems. Topologically, I will, in particular, indicate that the closures of the corresponding limit sets have the same, highly non-trivial, homology groups, in fact isomorphic to the homology groups of Hawaiian earrings, and that all these closures contain homeomorphic copies of Wazewski's universal dendrite.