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

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journal contribution
posted on 24.09.2019 by A. Sorrentino, Alexander Veselov
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].

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Nonlinearity

Volume

32

Issue

6

Pages

2147-2156

Citation

SORRENTINO, A. and VESELOV, A.P., 2019. Markov numbers, Mather's beta-function and stable norm. Nonlinearity, 32(6), 2147-2156

Publisher

London Mathematical Society

Version

AM (Accepted Manuscript)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

Acceptance date

05/02/2019

Publication date

2019-05-08

ISSN

0951-7715

Language

en

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