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

TL;DR

This paper explores the connections between Markov numbers, Mather's beta function, and the stable norm, revealing differentiability properties of a related function based on geometric and dynamical systems analysis.

Contribution

It establishes the relation between Fock's function and geometric invariants, proving differentiability at irrationals and non-differentiability at rationals using hyperbolic geometry and geodesic length results.

Findings

Proves differentiability of the function at irrational points.

Shows non-differentiability at rational points.

Links Markov numbers to geometric invariants and dynamical systems.

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].