### Dimension Group and Zero Dimensional Dynamical Systems

Ergodic Theory and Dynamical Systems Seminar

9th November 2023, 2:00 pm – 3:00 pm

Fry Building, G07

The first application of dimension group to zero-dimensional dynamical systems was in the work of I. Putnam in 1989 where he used dimension group to study interval exchange transformations (IET). Then in his works with T. Giordano, R. I. Herman, and C.F. Skau they developed those ideas which was based on creating Kakutani-Rokhlin (K-R) partitions for IET's to the general case of Cantor systems. They made a break through with this theory when they proved that every uniquely ergodic Cantor minimal system is orbit equivalent to either a Denjoy's or an odometer system; a topological analogues of the well-known Krieger's theorem in ergodic theory. Having sequences of K-R partitions, which is the bridge to dimension group, has been established for every zero-dimensional systems (by the works of S. Bezuglyi, K. Medynets, T. Downarowicz, O. Karpel and T. Shimomura) and is a strong tool in studying continuous and measurable spectrum of Cantor systems as well as topological factoring between them. In this talk after making an introduction to the notion of dimension group I will discuss some recent results about applications of that in studying spectrum and topological factoring of zero-dimensional systems.

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