Abstract: The visit sums and averages of orbits under irrational
rotations has been of interest since at least the seminal work of Kesten
in the 1960.
Recent work by Avila et al. revisted the problem in the case that the
rotation number is a quadratic irrational, showing that the typical
behaviour is slightly different from what Kesten proved for typical
rotation numbers. Their method is based on $\Z$-extension
(skew-products) over the rotation and certain renormalization
techniques.
We recently generalized their method so as to include samples of
Ehrenfest wind-tree model
as well.

Using purely combinatoric methods, it is possible to give precise
statements about visit sums for particular orbits, or actually the orbit
of 0. This orbit is actually non-typical in view to the results of Avila
et al.

The talk is based on joint work with Charles Fourgeron, Davide Ravotti,
Dalia Terhesiu, and another with Robert Fokkink.