BGU Probability and Ergodic Theory (PET) seminar
IcalAtom

The seminar meets on Thursdays, 11:00-12:00, in -101

This Week

Yiftach Dayan (Tel-Aviv University)

Diophantine approximations on random fractals

We will present a model for construction of random fractals which is called fractal percolation. The main 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. An 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 sketch the proof of this theorem and present 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.

We will present a model for construction of random fractals which is called fractal percolation. The main 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. An 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 sketch the proof of this theorem and present 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.

Chandrika Sadanand (The Hebrew University of Jerusalem)

Consider a polygon-shaped billiard table on which a ball can roll along straight lines and reflect off of edges infinitely. In work joint with Moon Duchin, Viveka Erlandsson and Chris Leininger, we have characterized the relationship between the shape of a polygonal billiard table and the set of possible infinite edge itineraries of balls travelling on it. In this talk, we will explore this relationship and the tools used in our characterization (notably a new rigidity result for flat cone metrics).

We will consider a class of groups defined by their action on Cantor space and use the invariant of finiteness properties to find among these groups an infinite family of quasi-isometry classes of finitely presented simple groups.

This is a joint work with Stefan Witzel and Matthew C. B. Zaremsky.

We introduce the notion of stationary actions in the context of C-algebras, and prove a new characterization of C-simplicity in terms of unique stationarity. This ergodic theoretical characterization provides an intrinsic understanding for the relation between C-simplicity and the unique trace property, and provides a framework in which C-simplicity and random walks interact. Joint work with Mehrdad Kalantar.

We apply the Tur\´an sieve and the simple sieve developed by Ram Murty and Yu-Ru Liu to study problems in random graph theory. More speciﬁcally, we obtain bounds on the probability of a graph having diameter 2 (or diameter 3 in the case of bipartite graphs). An interesting feature revealed in these results is that the Tur´an sieve and the simple sieve “almost completely” complement to each other. This is joint work with Yu-Ru Liu.

The talk discusses a convexity structure on metric spaces which
we call sheltered sets. This structure arises in the study
of the dynamics of the maximum cellular automaton over the binary alphabet
on finitely generated groups. I will discuss relations to
horoballs and dead ends in groups and present many open questions.
This is work in progress with Tom Meyerovitch.