Markov numbers, Mather's beta-function and stable norm
journal contributionposted on 24.09.2019 by A. Sorrentino, Alexander Veselov
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V. Fock  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  and McShane-Rivin .
- Mathematical Sciences