Flat surfaces, Bratteli diagrams, and unique ergodicity \`a la Masur

TL;DR

This paper links Bratteli diagrams and flat surface dynamics, establishing a Masur-like criterion for unique ergodicity based on the behavior of a renormalizing shift map.

Contribution

It introduces a novel criterion for unique ergodicity of flat surface flows using the dynamics of Bratteli diagrams and their shift maps.

Findings

A Masur-like criterion for unique ergodicity is established.

The shift map dynamics serve as a renormalization tool for flat surfaces.

Conditions for unique ergodicity based on accumulation points are identified.

Abstract

Recalling the construction of a flat surface from a Bratteli diagram, this paper considers the dynamics of the shift map on the space of all bi-infinite Bratteli diagrams as the renormalizing dynamics on a moduli space of flat surfaces of finite area. A criterion of unique ergodicity similar to that of Masur's for flat surface holds: if there is a subsequence of the renormalizing dynamical system which has a good accumulation point, the translation flow or Bratteli-Vershik transformation is uniquely ergodic. Related questions are explored.