Ergodic theory and diophantine approximation for translation surfaces and linear forms

Jayadev S. Athreya, Andrew Parrish, Jimmy Tseng

Research output: Contribution to journalArticle (Academic Journal)peer-review

5 Citations (Scopus)
237 Downloads (Pure)

Abstract

We derive results on the distribution of directions of saddle connections on translation surfaces using only the Birkhoff ergodic theorem applied to the geodesic flow on the moduli space of translation surfaces. Our techniques, together with an approximation argument, also give an alternative proof of a weak version of a classical theorem in multi-dimensional Diophantine approximation due to W. Schmidt [24, 25]. The approximation argument allows us to deduce the Birkhoff genericity of almost all lattices in a certain submanifold of the space of unimodular lattices from the Birkhoff genericity of almost all lattices in the whole space and similarly for the space of affine unimodular lattices.
Original languageEnglish
Pages (from-to)2173–2190
Number of pages18
JournalNonlinearity
Volume29
Issue number8
Early online date28 Jun 2016
DOIs
Publication statusPublished - Aug 2016

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