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.