Logarithmic laws and unique ergodicity
Jon Chaika Rodrigo Treviño

We show that Masur's logarithmic law of geodesics in the moduli space of translation surfaces does not imply unique ergodicity of the translation flow, but that a similar law involving the flat systole of a Teichmüller geodesic does imply unique ergodicity. It shows that the flat geometry has a better control on ergodic properties of translation flow than hyperbolic geometry.

keywords: Unique ergodicity translation surfaces logarithm laws
Infinite type flat surface models of ergodic systems
Kathryn Lindsey Rodrigo Treviño
We propose a general framework for constructing and describing infinite type flat surfaces of finite area. Using this method, we characterize the range of dynamical behaviors possible for the vertical translation flows on such flat surfaces. We prove a sufficient condition for ergodicity of this flow and apply the condition to several examples. We present specific examples of infinite type flat surfaces on which the translation flow exhibits dynamical phenomena not realizable by translation flows on finite type flat surfaces.
keywords: ergodic renormalization Bratteli diagram Translation surface cutting and stacking dictionary. Teichmüller dynamics odometer

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