Given a Lagrangian L, Mather introduced the $\beta$-function of L, which is a convex function. Many interesting properties of the Euler–Lagrange flow can be derived from the study of the behaviour of the $\beta$-function. In this work we obtain some links between the vertices of the $\beta$-function and the Hausdorff dimension of the associated invariant measures, from which we recover a result of differentiability of the $\beta$-function.