Bratteli diagrams, translation flows and their $C^*$-algebras

TL;DR

This paper explores the connections between Bratteli diagrams, translation surfaces, and their associated $C^*$-algebras, providing explicit links and K-theory relations, especially for finite genus surfaces.

Contribution

It establishes explicit topological and algebraic links between Bratteli diagrams, translation surfaces, and their $C^*$-algebras, extending previous combinatorial constructions.

Findings

Linked path spaces to surfaces explicitly

Related $C^*$-algebras and foliation structures

Analyzed finite genus surfaces with Rauzy-Veech induction

Abstract

In [LT16], Kathryn Lindsey and the second author constructed a translation surface from a bi-infinite Bratteli diagram. We continue an investigation into these surfaces. The construction given in [LT16] was essentially combinatorial. Here, we provide explicit links between the path space of the Bratteli diagram and the surface, including various intermediate topological spaces. This allows us to relate the $C^{*}$-algebras associated with tail equivalence on the Bratteli diagram and the foliation of the surface, under some mild hypotheses. This also allows us to relate the K-theory of the $C^{*}$-algebras involved. We also treat the case of finite genus surfaces in some detail, where the process of Rauzy-Veech induction (and its inverse) provide an explicit construction of the Bratteli diagrams involved.