Markov numbers, Mather’s β function and stable norm
Markov numbers, Mather’s β function and stable norm
复制标题
马尔可夫数、马瑟 β 函数和稳定范数
作者：
A. Sorrentino;A. Veselov
Fock（1997（arXiv：dg-ga/9702018 v3）; Fock et al 2007 Handbook of Teichmüller Theory（Zürich：European Mathematical Society））介绍了一个与马尔可夫数相关的有趣函数。我们解释了它与Federerer-Gromov稳定范数和Mather-函数的关系，并以此来研究它的性质。我们证明了，和它的自然推广是可微的，在任何无理x和不可微的，否则，通过利用与长度的关系，简单的封闭测地线上的穿孔或单孔环面与双曲度量和结果由Bangert（1994微积分变分偏微分。当量ag 2 49-63）和McShane-Rivin（1995 C. R. Acad. Sci.巴黎I 320）。
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).