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 subexponential complexity the Rauzy graphs converge to the line $\mathbf{Z}$ in the sense of Benjamini-Schramm convergence if and only if its complexity function $p(n)$ is unbounded and satisfies $\lim_n\frac{p(n+1)}{p(n)} = 1$. We then apply this criterion to many examples of well-studied dynamical systems. If the subshift is moreover uniquely ergodic then we show that the limit of labelled Rauzy graphs if it exists can be identified with the unique invariant measure. In addition we consider an example of a non uniquely ergodic system recently studied by Cassaigne and Kabor\'e and identify a continuum of invariant measures with subsequential limits of labelled Rauzy graphs.