Schedule

The cost of a probability measure preserving action of a countable group G on X is an invariant that generalizes the rank (minimal number of generators) of G and measures the “minimal average number of maps” needed to connect every pair of points of X in the same G orbit. This invariant rose to prominence after it was used by Gaboriau to distinguish the orbit equivalence relations of essentially free actions of two free groups on different number of generators. The fixed price conjecture predicts that any two essentially free p.m.p. actions of the group G have the same cost. In my talk I will report on a joint work with Sam Mellick and Amanda Wilkens in which we prove fixed price one for higher ranks lattices in semisimple real or p-adic groups. As a corollary we obtain that the number of generators of index n subgroup of such a group grows like o(n) which implies new state of the art results on the growth of mod-p homology groups. The proof is based on certain miraculous properties of Poisson-Voronoi tessellations of higher rank symmetric spaces that might be of independent interest.

Following Birkhoff's proof of the Pointwise Ergodic Theorem, it has been well studied whether convergence still holds along various subsequences. In 2020, Bergelson and Richter showed that under the additional assumption of unique ergodicity, pointwise convergence holds along the number theoretic sequence Omega(n), where Omega(n) denotes the number of prime factors of n counted with multiplicity. In this talk, we will see that, removing this assumption, a pointwise ergodic theorem does not hold along Omega(n). Time permitting, we will then study the interplay of the dynamics with certain number theoretic properties of Omega(n) to obtain further information on the asymptotic behavior of this sequence.

We will consider dynamical systems that we call Cantor repellers which are expanding maps on invariant Cantor sets coming from iterated function systems. Cantor repellers have two natural invariant measures: the measure of full dimension and the measure of maximal entropy. We show that dimensions and Lyapunov exponents of those measures are flexible up to well understood restrictions. We will also discuss the boundary case for the range of values of the considered dynamical data. This is joint work with Jacob Mazor.

I will discuss the dynamics of the horocycle slow on a stratum of translation surfaces (which is an invariant subvariety of the bundle Omega M_g of holomorphic one forms over the moduli space of genus g Riemann surfaces). This flow can be defined as the action of upper triangular matrices with eigenvalue 1, acting linearly on flat charts. Work of Ratner on unipotent flows on homogeneous spaces leads to the question of whether the orbit-closures and invariant measures for this action can be meaningfully classified. I will quickly survey both positive and negative results in this direction – in particular, a new result on orbit-closure classification in rank one loci. The talk will be based on joint work with Bainbridge, Chaika, Smillie, and Ygouf (in various combinations).