Modul: MAT760 Ergodic Theory and Dynamical Systems Seminar

## Morning Seminar: Diophantine approximations on random fractals

Yiftach Dayan talk

**Date:** 19.10.18 **Time:** 09.00 - 10.00 **Room:** Y27H28

We will present a model for the construction of random fractals which is called fractal percolation. The result that will be presented in this talk states that a typical fractal percolation set E intersects every set which is winning for a certain game that is called the "hyperplane absolute game", and the intersection has the same Hausdorff dimension as E. A motivating example of such a winning set is the set of badly approximable vectors in dimension d. In order to prove this theorem one may show that a typical fractal percolation set E contains a sequence of Ahlfors-regular subsets with dimensions approaching the dimension of E, where all the subsets in this sequence are also "hyperplane diffuse", which means that they are not concentrated around affine hyperplanes when viewed in small enough scales. If time permits, we will discuss the method of the proof of this theorem as well as a generalization to a more general model for random construction of fractals which is given by projecting Galton-Watson trees against any similarity IFS whose attractor is not contained in a single affine hyperplane.