Dynamical generalizations of the lagrange spectrum

We compute two invariants of topological conjugacy, the upper and lower limits of the inverse of Boshernitzan??s ne _n , where e _n is the smallest measure of a cylinder of length n, for three families of symbolic systems: the natural codings of rotations, three-interval exchanges, and Arnoux-Rauzy systems. The sets of values of these invariants for a given family of systems generalize the Lagrange spectrum, which is what we obtain for the family of rotations with the upper limit of 1/ne _n.