Geodesic laminations as geometric realizations of Arnoux--Rauzy sequences

Abstract

We consider a family of minimal sequences on a $3$-symbol alphabet with complexity $2n+1$, which satisfy a special combinatorial property. These sequences were originally defined by P. Arnoux and G. Rauzy as a generalization of the binary sturmian sequences. We prove that the dynamical system associated to each of these sequences of this family, can be realized as a dynamical system defined on a geodesic lamination on the hyperbolic disk. This is a generalization of the results shown in a previous paper of the author. We also show some applications of these results.