Flat surfaces, Bratteli diagrams and unique ergodicity à la Masur (Q1650028)

Relying on the fact that a weighted, fully ordered bi-infinite Bratteli diagram determines a flat surface surface of area 1 [\textit{K. Lindsey} and the author, Discrete Contin. Dyn. Syst. 36, No. 10, 5509--5553 (2016; Zbl 1366.37105)] and motivated by \textit{Masur's criterion}, which establishes that the Teichmüller orbit of a flat surface is divergent if the translation flow is uniquely ergodic, the author studies the dynamics of the shift map on the space of bi-infinite Bratteli diagrams in order to understand translation flows. The main result of the paper is Theorem~1, which states that if the orbit of some Bratteli diagram \(\mathcal{B}\) has an accumulation point which is minimal, then \(\mathcal{B}\) determines a unique flat surface \(S\) such that the vertical flow on \(S\) is uniquely ergodic. In addition, the assumption about the existence of an accumulation point only is not sufficient to ensure ergodicity.