Limit of Rauzy graphs

To any language $L \subseteq A^*$ one can naturally associate an infinite family of finite graphs $R_L(n)$, called the Rauzy graphs of $L$. Vertices of $R_L(n)$ are words of length n in L, while edges represent overlaps and are words of length n+1. If $L=A^*$, then the corresponding Rauzy graphs are well-known under the name of de Bruijn graphs. Since we have an infinite family, it is natural to ask if the limit of the $R_L(n)$ exists, and if yes if it can be computed from L. In this talk, we will answer this question for L=A^*, for languages of subexponential complexity, and partially for subshifts of finite type. In each case, the limit can be expressed as an horocyclic product of trees, which can be explicitly computed from L. This is partially joint work with Rostislav Grigorchuk, Tatiana Nagnibeda, Alexandra Skripchenko and Georgii Veprev.