PSU Mark
Eberly College of Science Mathematics Department

Meeting Details

For more information about this meeting, contact Anatole Katok, Mari Royer, Yakov Pesin, Dmitri Burago.

Title:The irregular set for the beta transformation has full topological entropy (= log beta) and full Hausdorff dimension (=1)
Seminar:Center for Dynamics and Geometry Seminars
Speaker:Daniel Thompson, Penn State
Abstract:
A recent weakening of the specification property provides new tools to study interesting systems beyond the scope of the standard theory of uniformly hyperbolic dynamics such as the beta-transformation. This property was introduced by Pfister and Sullivan as the g-almost product property. The version we study is a priori slightly weaker and we rename it the almost specification property. We show that for dynamical systems with almost specification, the set of points for which the Birkhoff average of a continuous function does not exist (which we call the irregular set) is either empty or has full topological entropy. Every beta-shift satisfies almost specification and we show that the irregular set for any beta-shift or beta-transformation is either empty or has full topological entropy and Hausdorff dimension. The talk is in three parts. Firstly, we give some history on results about the topological entropy and Hausdorff dimension of the irregular set. Secondly, we discuss our abstract results and try to give some intuition as to why they are true. Thirdly, we discuss in detail the application to the beta-transformation. In particular, we give some intuition on the almost specification property and show why it is satisfied by the beta-transformation.

Room Reservation Information

Room Number:MB106
Date:10 / 12 / 2009
Time:03:30pm - 05:30pm