Title: Unique ergodicity of branched covers of translation surfaces
Abstract: We will start by introducing translation surfaces — flat surfaces with cone singularities and straight-line flow. These are among the simplest examples of dynamical systems, yet they model a variety of physical processes, such as Ehrenfest wind-tree models, polygonal billiards, optical cavities, and Eaton lenses. Translation surfaces are chaotic, or, more specifically, uniquely ergodic in almost every direction: for almost every initial point, the straight trajectory equidistributes for area, and time averages equal space averages (this idea comes from Boltzmann's ergodic hypothesis in thermodynamics). After a primer where I will define everything we need, I will present a new construction on translation surfaces called branched n-covers: on a uniquely ergodic translation surface X, pick a slit s=[P, Q]; take n copies and switch sheets i → i+1 mod n each time the vertical flow hits the slit s (i.e., we glue the copies together). It turns out that the unique ergodicity property is robust for such covers under fairly weak constraints. Moreover, the conditions are geometric despite the measure-theoretic core of the problem. This is especially notable because the setup is starkly different from the standard one in this area, where the varied parameter is not the new surface construction but the direction of the flow. (Joint with Elizaveta Shuvaeva.)
Bryna Kra
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