Markov Numbers, Mather's $\beta$ function and stable norm

Abstract

V. Fock [7] introduced an interesting function $\psi (x)$, $x\in \mathbb{R}$ related to Markov numbers. We explain its relation to Federer-Gromov's stable norm and Mather's $\beta$-function, and use this to study its properties. We prove that $\psi$ and its natural generalisations are differentiable at every irrational $x$ and non-differentiable otherwise, by exploiting the relation with length of closed geodesics on the punctured or one-hole tori with the hyperbolic metric and the results by Bangert [3] and McShane- Rivin [19].

9 pages, 2 figures

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