##
**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.

Reviewer: Michel Waldschmidt (Paris)

### MSC:

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.) |

### References:

[1] | Bugeaud, Yann, Sets of exact approximation order by rational numbers. {II}, Unif. Distrib. Theory. Uniform Distribution Theory, 3, 9-20, (2008) · Zbl 1212.11071 |

[2] | Cusick, Thomas W.; Flahive, Mary E., The {M}arkoff and {L}agrange {S}pectra, Math. Surveys Monogr., 30, x+97 pp., (1989) · Zbl 0685.10023 |

[3] | Falconer, K. J., The Geometry of Fractal Sets, Cambridge Tracts in Math., 85, xiv+162 pp., (1986) |

[4] | Ferenczi, S.; Mauduit, C.; Moreira, C. G., An algorithm for the word entropy, (2018) · Zbl 1398.68415 |

[5] | Fre{\u{\i}}man, G. A., {D}iofantovy Priblizheniya i Geometriya Chisel (Zadacha {M}arkova), 144 pp., (1975) · Zbl 0347.10025 |

[6] | Hall, Jr., Marshall, On the sum and product of continued fractions, Ann. of Math. (2). Annals of Mathematics. Second Series, 48, 966-993, (1947) · Zbl 0030.02201 |

[7] | Jarn{\'\i k}, V., Zur metrischen {T}heorie der diophantischen {A}pproximationen, Prace Mat.-Fiz., 36, 91-106, (1929) · JFM 55.0718.01 |

[8] | Markoff, A., Sur les formes quadratiques binaires ind\'efinies, Math. Ann.. Mathematische Annalen, 17, 379-399, (1880) · JFM 12.0143.02 |

[9] | Matheus, C.; Moreira, C. G., \({HD(M\setminus L)} > 0.353\), (2017) · Zbl 1440.11121 |

[10] | Matheus, C.; Moreira, C. G., Markov spectrum near {F}reiman’s isolated points in \({M}\setminus{L}\), (2018) |

[11] | Matheus, C.; Moreira, C. G., New numbers in \({M}\setminus{L}\) beyond \(\sqrt{12}\): solution to a conjecture of {C}usick, (2018) |

[12] | Moreira, C. G., Geometric properties of images of cartesian products of regular {C}antor sets by differentiable real maps, (2016) |

[13] | Moreira, Carlos Gustavo; Yoccoz, Jean-Christophe, Stable intersections of regular {C}antor sets with large {H}ausdorff dimensions, Ann. of Math. (2). Annals of Mathematics. Second Series, 154, 45-96, (2001) · Zbl 1195.37015 |

[14] | Perron, O., {\"{U}}ber die Approximation irrationaler {Z}ahlen durch rationale {II}, S.-B. Heidelberg Akad. Wiss., 8, 12 pp. pp., (1921) · JFM 48.0193.01 |

[15] | Palis, Jacob; Takens, Floris, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Fractal Dimensions and Infinitely Many Attractors, Cambridge Stud. Adv. Math., 35, x+234 pp., (1993) · Zbl 0790.58014 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.