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Dynamical characterization of initial segments of the Markov and Lagrange spectra

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Abstract

We prove that, for every \(k\ge 4\), the sets M(k) and L(k), which are Markov and Lagrange dynamical spectra related to conservative horseshoes and associated to continued fractions with coefficients bounded by k coincide with the intersections of the classical Markov and Lagrange spectra with \((-\infty ,\sqrt{k^2+4k}]\). We also observe that, despite the corresponding statement is also true for \(k=2\), it is false for \(k=3\).

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References

  1. Astels, S.: Cantor sets and numbers with restricted partial quotients, Ph.D. thesis, Univerity of Waterloo (1999)

  2. Arnoux, P.: Le codage du flot géodésique sur la surface modulaire (2). Enseign. Math. 40(1–2), 29–48 (1994)

    MathSciNet  MATH  Google Scholar 

  3. Cusick, T., Flahive, M.: The Markoff and Lagrange spectra, Mathematical Surveys and Monographs, vol. 30. American Mathematical Society, Providence, RI (1989)

    Book  Google Scholar 

  4. Cerqueira, A., Matheus, C., Moreira, C.G.: Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra. J. Mod. Dyn. (2018). https://doi.org/10.3934/jmd.2018006

    Article  MathSciNet  MATH  Google Scholar 

  5. Freiman, G.A.: Diophantine Approximation and Geometry of Numbers (The Markoff Spectrum). Kalininskii Gosudarstvennyi Universitet, Moscow (1975)

    Google Scholar 

  6. Hall, M.: On the sum and product of continued fractions. Ann. Math. 48, 966–993 (1947)

    Article  MathSciNet  Google Scholar 

  7. Ito, S.: Number theoretic expansions, Algorithms and Metrical observations. Séminaire de Théorie des Nombres de Bordeaux, 1–27 (1984)

  8. Lima, D., Moreira, C.G.: Phase transitions on the Markov and Lagrange dynamical spectra. Annales de l’Institut Henri Poincaré C Analyse Non Linéaire 38(5), 1429–1459 (2021). https://doi.org/10.1016/j.anihpc.2020.11.007

    Article  MathSciNet  MATH  Google Scholar 

  9. Lima, D., Matheus, C., Moreira, C.G., Romaña, S.: Classical and Dynamical Markov and Lagrange spectra. World Scientific, Singapore (2020)

    Book  Google Scholar 

  10. Markov, A.: Sur les formes quadratiques binaires indéfinies. Math. Ann. 15, 381–406 (1879)

    Article  Google Scholar 

  11. Moreira, C.G.: Sums of regular Cantor sets, dynamics and applications to number theory. Period. Math. Hung. 37(1), 55–63 (1998)

    Article  MathSciNet  Google Scholar 

  12. Moreira, C.G.: Conjuntos de Cantor, Dinâmica e Aritmética, 22o. Colóquio Brasileiro de Matemática (1999)

  13. Moreira, C.G.: Geometric properties of the Markov and Lagrange spectra. Ann. Math. 188(1), 145–170 (2018)

    Article  MathSciNet  Google Scholar 

  14. Moreira, C.G.: On the minima of Markov and Lagrange Dynamical Spectra, Astérisque No. 415, Quelques aspects de la théorie des systèmes dynamiques: un hommage à Jean-Christophe Yoccoz. I, 45–57 (2020)

  15. Matheus, C., Moreira, C.G., Pollicott, M., Vytnova, P.: Hausdorff Dimension of Gauss–Cantor Sets and Two Applications to Classical Lagrange and Markov Spectra. https://arxiv.org/abs/2106.06572

  16. Moreira, C.G., Romaña, S.: On the Markov and Lagrange dynamical spectra. Ergod. Theory Dyn. Syst. 37(5), 1570–1591 (2017). https://doi.org/10.1017/etds.2015.121

    Article  MATH  Google Scholar 

  17. Perron,O.: Über die Approximation irrationaler Zahlen durch rationale II, S.-B. Heidelberg Akad. Wiss., Abh. 8 (1921)

  18. Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73(6), 747–817 (1967)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The first author is partially supported by CNPq by the project Alagoas Dinâmica - 409198/2021-8, CNPq/MCTI/FNDCT No 18/2021 - Faixa A and by FAPEAL. We also thank the anonymous referee for his valuable suggestions.

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Author notes

  1. Davi Lima

    Present address: Instituto de Matemática, Universidade Federal de Alagoas, Av Lourival Melo Mota s/n, Maceió, AL, Brazil

  2. Carlos Gustavo Moreira

    Present address: Instituto Nacional de Matemática Pura e Aplicada, IMPA, Estrada Dona Castorina, 110, Rio de Janeiro, RJ, Brazil

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    1. Davi Lima
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    2. Carlos Gustavo Moreira
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    Correspondence to Davi Lima.

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    Davi Lima partially supported by CNPq and FAPEAL.

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    Lima, D., Moreira, C.G. Dynamical characterization of initial segments of the Markov and Lagrange spectra. Monatsh Math 199, 817–852 (2022). https://doi.org/10.1007/s00605-022-01765-3

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