### Equidistribution of divergent orbits in the space of lattices

Ergodic Theory and Dynamical Systems Seminar

1st March 2018, 3:00 pm – 4:00 pm

Howard House, 4th Floor Seminar Room

A well known application of the pointwise ergodic theorem (PET) states that almost every x in [0,1] has a normal continued fraction expansion, or equivalently its orbit under the Gauss map T(x):={1/x} (where {y} is the fractional part of y) equidistributes with respect to the Gauss Kuzmin measure dt/( ln(2)(1+t) ). This is not true for all x, and in particular it fails for rational numbers which have finite continued fraction expansions.

In this talk we shall see how to "extend" the PET to rational numbers and its connection to divergent orbits of the diagonal group in the space of 2-dimensional lattices. Furthermore, we shall show how the natural setting of this problem is actually over the adeles, and in particular it can be formulated in any dimension (for which we give some partial results).

This is a joint work with Uri Shapira from the Technion.

## Comments are closed.