Markov numbers, Mather’s beta function and stable norm

Fock (1997 (arXiv:dg-ga/9702018v3); Fock et al 2007 Handbook of Teichmüller Theory (Zürich: European Mathematical Society)) introduced an interesting function related to Markov numbers. We explain its relation to Federer–Gromov's stable norm and Mather's -function, and use this to study its properties. We prove that 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 one-holed tori with the hyperbolic metric and the results by Bangert (1994 Calculus Variations Partial Differ. Equ. 2 49–63) and McShane–Rivin (1995 C. R. Acad. Sci. Paris I 320).