Logarithmic laws and unique ergodicity

TL;DR

The paper demonstrates that a logarithmic law involving flat systole in Teichmüller geodesics guarantees unique ergodicity of translation flows, highlighting the superior control of flat geometry over hyperbolic geometry for ergodic properties.

Contribution

It establishes that a logarithmic law based on flat systole implies unique ergodicity, contrasting with Masur's law in hyperbolic geometry which does not.

Findings

Logarithmic law involving flat systole implies unique ergodicity.

Masur's hyperbolic law does not imply unique ergodicity.

Flat geometry provides better control over ergodic properties.

Abstract

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 Teichm\"{u}ller geodesic does imply unique ergodicity. It shows that the flat geometry has a better control on ergodic properties of translation flow than hyperbolic geometry.