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Groups And Dynamics
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Jon Chaika, Zoom: A strange limit of horocycle ergodic measures in a stratum of translation surfaces (part 1)
Thursday, October 21, 2021, 11:00am - 12:00pm
This two part talk takes the perspective that ergodic theory is a subarea of dynamical systems that uses measures to understand the long term behavior of points under iterates of a function from a space to itself. The first part of this talk will motivate this perspective by discussing general results, like the Birkhoff ergodic theorem and the Krylov-Bogoliobov Theorem, and the case of special dynamical systems like, irrational rotations of the circle, the full shift on two symbols and the horocycle flow on translation surfaces. The second part of the talk will introduce the space of translation surfaces SL(2,R)'s action on this space. From here it will state the main result of the talk: in the space of unit area genus two translation surfaces with one cone point, there is a weak-star limit of measures on periodic horocycles that is fully supported in the 7-dimensional space but gives positive measure to a 3-dimensional submanifold. As a consequence, we obtain a non-genericity result for the horocycle flow in this space. If time permits, I will discuss a bit of the idea of the proof, that inhomogeneity in the derivative cocycle along a 1-periodic horocycle produces long periodic horocycles with this perhaps strange behavior. This is joint work with Osama Khalil and John Smillie.
Location: Zoom

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