Posted by: yglima | May 28, 2010

## Z^d-actions with prescribed topological and ergodic properties

Last Moday, I gave a talk at the Bicentennial Workshop of Dynamical Systems, which has been held at the beautiful city of San Pedro de Atacama, Chile. This was my first talk in a conference and was based on the work I did while I was at The Ohio State University last year, under the advising of Vitaly Bergelson. I would like to thank Vitaly and the Department of Mathematics of OSU for its great hospitality and possibility of the development of nice conditions of work and also Carlos Matheus for, once more, inviting me for writing in this blog, which is among the 50 best blogs for Math majors (see the ranking here).

The theorem is a topological counterpart to Bourgain´s theorem on the a.e. existence of ergodic averages along polynomials (see ERT3) for $\mathbb Z^d$-actions. It extends a previous work of one of Bergelson´s student, Ronnie Pavlov, who proved it for $\mathbb Z$-actions.

The idea is to construct an increasing sequence of finite alphabets/words with controlled combinatorial and statistical properties in such a way that we have freedom of combinatorics at a set which is negligible (in the sense of zero upper-Banach density) inside $\mathbb Z^d$.

As the slides are self-contained, it would be a loss of time for me and loss of patience for the reader to repeat everything, so I´m just making them available here. I hope you learn something with it!