Gordon: Barak Weiss (TAU) - Diophantine approximation on fractals and random walks on the space of lattices and its S-adic extension

Date: 
Tue, 31/12/202412:00
Location: 
Manchester lobby

 

Speaker: Barak Weiss (Tel Aviv University) 

Title:  Diophantine approximation on fractals and random walks on the space of lattices and its S-adic extension


Abstract:

Let ν be a Bernoulli measure on a fractal in Rd generated by a finite collection of contracting similarities with no rotations and with rational coefficients; for instance, the usual coin tossing measure on Cantor's middle thirds set. Let at = diag (et,..., et,e-dt), let U be its expanding horospherical group, which we identify with Rd, and let \bar ν be the pushforward of ν onto the space of lattices SL(d+1,R)/SL(d+1,Z), via the orbit map of the identity coset under U. In joint work in progress with Khalil and Luethi, we show that the pushforward of \bar ν under a_t equidistributes as t tends to infinity, as do the pushforwards under more general one parameter subgroups. This generalizes a previous result of Khalil and Luethi and implies that on a large class of rational self similar fractals, weighted badly approximately vectors are of zero measure. I will discuss these Diophantine applications and some probabilistic ideas used in the proof.