Seminar 3.9.20 Climenhaga

Title: Non-adapted measures for billiards and other systems with singularities

Speaker: Vaughn Climenhaga – University of Houston

Abstract: The extension of smooth ergodic theory to systems with singularities, such as billiards, generally requires one to work with “adapted” measures, which do not give too much weight to the neighborhoods of the singularities of the system. For hyperbolic systems such as the Sinai billiard, it is often the case that natural invariant measures, such as the SRB measure and the measure of maximal entropy (MME), are adapted. More generally one can ask about equilibrium measures, and it becomes important to understand how large the entropy of a non-adapted measure can be. I will describe some simple examples illustrating some of the possible behaviors for interval maps, as well as an example of a billiard system with a positive entropy non-adapted measure (joint work with Mark Demers, Yuri Lima, and Hongkun Zhang). Finally, I will formulate some conjectures and describe work in progress towards realizing them.