BGU Probability and Ergodic Theory (PET) seminar Ical Atom

For a minimal action of a countable group G on a compact space X, we establish necessary conditions for the simplicity of the corresponding reduced crossed product C*-algebras in terms of stabilizer subgroups. In particular, our result gives a complete characterization of the simplicity of the reduced crossed product associated with minimal actions of linear groups, answering a question of Ozawa (2014) for these groups. Joint work with Mehrdad Kalantar

In 1965, Wolfgang Schmidt introduced the $(\alpha,\beta)$-Schmidt game as a dynamical tool for studying fundamental sets in Diophantine approximation. In particular, he proved that in Hilbert spaces these games can be used to obtain lower bounds on the Hausdorff dimension of sets that are small from the measure-theoretic point of view but large in a fractal sense. Schmidt’s approach relies on the underlying geometry of the space.

In this talk, I will introduce these games and present an analogous result in the setting of complete doubling metric spaces, where we replace geometric arguments with a purely game-theoretic approach.

This is joint work with Itamar Bellaïche. No prior knowledge of game theory is assumed.

Let $k \geq 2$ be an integer. In 2000, Baake, Moody, and Pleasants proved that the set of lattice points in $\mathbb{Z}^k$ visible from the origin has pure point diffraction. It is also known that irreducible cut-and-project sets—such as the Ammann-Beenker point set—exhibit pure point diffraction.

Let $S$ be a finite subset of $\mathbb{Z}^k$, and let $V(S)$ be the set of points simultaneously visible from $S$. We will discuss the diffraction spectrum of the set $V(S)$ and the diffraction spectrum of the set of visibility from the origin in certain classes of irreducible cut-and-project sets. Joint work with Carlos Ospina.