The Lagrange and Markov Spectra from the Dynamical Point of View

  • Carlos Matheus
Part of the Lecture Notes in Mathematics book series (LNM, volume 2213)


This text grew out of my lecture notes for a 4-h minicourse delivered on October 17 and 19, 2016 during the research school “Applications of Ergodic Theory in Number Theory”—an activity related to the Jean-Molet Chair project of Mariusz Lemańczyk and Sébastien Ferenczi—realized at CIRM, Marseille, France. The subject of this text is the same as my minicourse, namely, the structure of the so-called Lagrange and Markov spectra (with a special emphasis on a recent theorem of C.G. Moreira).


  1. 1.
    P. Arnoux, Le codage du flot géodésique sur la surface modulaire. Enseign. Math. (2) 40(1–2), 29–48 (1994)Google Scholar
  2. 2.
    A. Cerqueira, C. Matheus, C.G. Moreira, Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra. Preprint (2016) available at arXiv:1602.04649Google Scholar
  3. 3.
    T. Cusick, M. Flahive, The Markoff and Lagrange Spectra. Mathematical Surveys and Monographs, vol. 30 (American Mathematical Society, Providence, RI, 1989), x+97 pp.Google Scholar
  4. 4.
    P.G. Dirichlet, Verallgemeinerung eines Satzes aus der Lehre von den Kettenbrüchen nebst einigen Anwendungen auf die Theorie der Zahlen. pp. 633–638 Bericht über die Verhandlungen der Königlich Preussischen Akademie der Wissenschaften. Jahrg. 1842, S. 93–95Google Scholar
  5. 5.
    K. Falconer, The Geometry of Fractal Sets. Cambridge Tracts in Mathematics, vol. 85 (Cambridge University Press, Cambridge, 1986), xiv+162 pp.Google Scholar
  6. 6.
    G. Freiman, Non-coincidence of the spectra of Markov and of Lagrange. Mat. Zametki 3, 195–200 (1968)MathSciNetGoogle Scholar
  7. 7.
    G. Freiman, Non-coincidence of the spectra of Markov and of Lagrange, in Number-Theoretic Studies in the Markov Spectrum and in the Structural Theory of Set Addition (Russian) (Kalinin. Gos. Univ, Moscow, 1973), pp. 10–15, 121–125Google Scholar
  8. 8.
    G. Freiman, Diophantine Approximations and the Geometry of Numbers (Markov’s Problem) (Kalininskii Gosudarstvennyi University, Kalinin, 1975), 144 pp.zbMATHGoogle Scholar
  9. 9.
    M. Hall, On the sum and product of continued fractions. Ann. Math. (2) 48, 966–993 (1947)MathSciNetCrossRefGoogle Scholar
  10. 10.
    D. Hensley, Continued fraction Cantor sets, Hausdorff dimension, and functional analysis. J. Number Theory 40(3), 336–358 (1992)MathSciNetCrossRefGoogle Scholar
  11. 11.
    P. Hubert, L. Marchese, C. Ulcigrai, Lagrange spectra in Teichmüller dynamics via renormalization. Geom. Funct. Anal. 25(1), 180–255 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    A. Hurwitz, Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche, Math. Ann. 39(2), 279–284 (1891)MathSciNetCrossRefGoogle Scholar
  13. 13.
    O. Jenkinson, M. Pollicott, Computing the dimension of dynamically defined sets: E2 and bounded continued fractions. Ergodic Theory Dyn. Syst. 21(5), 1429–1445 (2001)CrossRefGoogle Scholar
  14. 14.
    O. Jenkinson, M. Pollicott, Rigorous effective bounds on the Hausdorff dimension of continued fraction Cantor sets: a hundred decimal digits for the dimension of E 2. Preprint (2016) available at arXiv:1611.09276Google Scholar
  15. 15.
    A. Khinchin, Continued Fractions (The University of Chicago Press, Chicago, London, 1964), xi+95 pp.Google Scholar
  16. 16.
    P. Lévy, Sur le développement en fraction continue d’un nombre choisi au hasard. Compos. Math. 3, 286–303 (1936)zbMATHGoogle Scholar
  17. 17.
    K. Mahler, On lattice points in n-dimensional star bodies. I. Existence theorems. Proc. R. Soc. Lond. Ser. A 187, 151–187 (1946)MathSciNetCrossRefGoogle Scholar
  18. 18.
    A. Markoff, Sur les formes quadratiques binaires indéfinies. Math. Ann. 17(3), 379–399 (1880)MathSciNetCrossRefGoogle Scholar
  19. 19.
    J. Marstrand, Some fundamental geometrical properties of plane sets of fractional dimensions. Proc. Lond. Math. Soc. (3) 4, 257–302 (1954)MathSciNetCrossRefGoogle Scholar
  20. 20.
    H. McCluskey, A. Manning, Hausdorff dimension for horseshoes. Ergodic Theory Dyn. Syst. 3(2), 251–260 (1983)MathSciNetCrossRefGoogle Scholar
  21. 21.
    C.G. Moreira, Geometric properties of the Markov and Lagrange spectra. Preprint (2016) available at arXiv:1612.05782Google Scholar
  22. 22.
    C.G. Moreira, Geometric properties of images of cartesian products of regular Cantor sets by differentiable real maps. Preprint (2016) available at arXiv:1611.00933Google Scholar
  23. 23.
    C.G. Moreira, S. Romaña, On the Lagrange and Markov dynamical spectra, Ergodic Theory Dyn. Syst. 1–22 (2016).
  24. 24.
    C.G. Moreira, J.-C. Yoccoz, Stable intersections of regular Cantor sets with large Hausdorff dimensions, Ann. Math. (2) 154(1), 45–96 (2001)Google Scholar
  25. 25.
    C.G. Moreira, J.-C. Yoccoz, Tangences homoclines stables pour des ensembles hyperboliques de grande dimension fractale. Ann. Sci. Éc. Norm. Supér. (4) 43(1), 1–68 (2010)MathSciNetCrossRefGoogle Scholar
  26. 26.
    S. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms. Inst. Hautes Études Sci. Publ. Math. 50, 101–151 (1979)MathSciNetCrossRefGoogle Scholar
  27. 27.
    J. Parkkonen, F. Paulin, Prescribing the behaviour of geodesics in negative curvature. Geom. Topol. 14(1), 277–392 (2010)MathSciNetCrossRefGoogle Scholar
  28. 28.
    D. Witte Morris, Ratner’s Theorems on Unipotent Flows. Chicago Lectures in Mathematics (University of Chicago Press, Chicago, IL, 2005), xii+203 pp.Google Scholar
  29. 29.
    D. Zagier, Eisenstein series and the Riemann zeta function, in Automorphic Forms, Representation Theory and Arithmetic (Bombay, 1979). Tata Institute of Fundamental Research Studies in Mathematics, vol. 10 (Tata Institute of Fundamental Research, Bombay, 1981), pp. 275–301CrossRefGoogle Scholar
  30. 30.
    D. Zagier, On the number of Markoff numbers below a given bound. Math. Comput. 39(160), 709–723 (1982)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Carlos Matheus
    • 1
  1. 1.Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS (UMR 7539)VilletaneuseFrance

Personalised recommendations