Dynamics & probability: Ofir David (HUJI), Equidistribution of finite continued fractions.

Tue, 29/11/201614:00-15:00
Manchester building, Hebrew University of Jerusalem, (Room 209)
It is well known that for almost every x in (0,1) its orbit under the Gauss map, namely T(x)=1/x-[1/x], equidistributes with respect to the Gauss-Kuzmin measure. This claim is not true for all x, and in particular it is not true for rational numbers which have finite "orbits" which terminate in 0. In order to still have some equidistribution, we instead group
together the orbits corresponding to p/q when q is fixed and (p,q)=1 and ask whether these finite sets equidistribute as q goes to infinity. 
In this talk I will discuss these finite orbits, and I will show how to formulate and solve this problem using the language of dynamics in SL_2(Z)\SL_2(R). This will in turn imply a stronger equidistribution of the continued fraction expansion of rational numbers.
This is a joint work with Uri Shapira.