Markov numbers, Mather's beta-function and stable norm

V. Fock [7] introduced an interesting function ψ(x), x ∈ R related to Markov numbers. We explain its relation to Federer-Gromov’s stable norm and Mather’s β-function, and use this to study its properties. We prove that ψ and its natural generalisations are differentiable at every irrational x and non-differentiable otherwise, by exploiting the relation with length of simple closed geodesics on the punctured or oneholed tori with the hyperbolic metric and the results by Bangert [3] and McShane-Rivin [23].