Limits of Rauzy graphs of languages with subexponential complexity

Abstract

To a subshift over a finite alphabet, one can naturally associate an infinite family of finite graphs, called its Rauzy graphs. We show that for a subshift of unbounded subexponential complexity, the Rauzy graphs converge in the sense of Benjamini–Schramm if and only if their empirical spectral measures converge, which is equivalent to the complexity function p(n) to satisfy \lim_{n}\frac{p(n+1)}{p(n)}= 1 . The limit in this case is the infinite line \mathbf{Z} . If the subshift is moreover uniquely ergodic, the limit of labelled Rauzy digraphs, if it exists, can be identified with the unique invariant measure. We also consider a non-uniquely ergodic example recently studied by Cassaigne and Kaboré and identify a continuum of invariant measures with subsequential limits of labelled Rauzy digraphs.