Geometric properties of the Markov and Lagrange spectra. (English) Zbl 1404.11095

Denote by \(L\) the Lagrange spectrum and by \(M\) the Markoff spectrum. Both are closed subsets of the half line \([\sqrt 5,\infty)\) and \(L\) is a subset of \(M\). The Hall ray \([c,\infty)\) is the biggest half line contained in \(L\) and it is the same for \(M\). While \(L\cap [\sqrt 5,3)=M\cap [\sqrt 5,3)\) consists of an increasing sequence with limit \(3\), the sets \(L\cap (3,c)\) and \(M\cap (3,c)\), which are distinct, are not so well known. The set \((M\setminus L)\cap [\sqrt 5,3)\) have recently been investigated in preprints from the a uthor and C. Matheus [“\(HD(M\setminus L)>0.353\)”, Preprint, arXiv:1703.04302; “Markov spectrum near Freiman’s isolated points in \(M\setminus L\)”, Preprint, arXiv:1802.02454; “New numbers in \(M\setminus L\) beyond \(\sqrt{12}\): solution to a conjecture of Cusick”, Preprint, arXiv:1803.01230]. In the paper under review, the author goes much further and proves a number of deep new results related with the Hausdorff dimensions of \(L\cap (-\infty,t)\) and \(M\cap (-\infty,t)\) for \(t\in{\mathbb{R}}\). In particular, he answers several questions raised by Y. Bugeaud [Unif. Distrib. Theory 3, No. 2, 9–20 (2008; Zbl 1212.11071)] on these sets. He proves that the Lagrange spectrum \(L\) is a perfect set: the derivative \(L'\) is equal to the second derivative \(L''\); the question whether \(M''=M'\) is yet unsolved. Several other open problems are stated. One main tool is from a joint work of the author with J.-C. Yoccoz [Ann. Math. (2) 154, No. 1, 45–96 (2001; Zbl 1195.37015)] on regular Cantor sets.


11J06 Markov and Lagrange spectra and generalizations
11J70 Continued fractions and generalizations
28A78 Hausdorff and packing measures
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
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