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Combinatorics on Words and the Theory of Markoff

  • Christophe ReutenauerEmail author
Conference paper
Part of the Abel Symposia book series (ABEL, volume 13)

Abstract

This is a survey on the theory of Markoff, in its two aspects: quadratic forms (the original point of view of Markoff), approximation of reals. A link wih combinatorics on words is shown, through the notion of Christoffel words and special palindromes, called central words. Markoff triples may be characterized, by using some linear representation of the free monoid, restricted to these words, and Fricke relations. A double iterated palindromization allows to construct all Markoff numbers and to reformulate the Markoff numbers injectivity conjecture (Frobenius, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin 26:458–487, 1913).

References

  1. 1.
    Aigner, M.: Markov’s Theorem and 100 Years of the Uniqueness Conjecture. Springer, Berlin (2013)CrossRefGoogle Scholar
  2. 2.
    Berstel, J., de Luca, A.: Sturmian words, Lyndon words and trees. Theor. Comput. Sci. 178, 171–203 (1997)CrossRefGoogle Scholar
  3. 3.
    Berstel, J., Lauve, A., Reutenauer, C., Saliola, F.: Combinatorics on Words: Christoffel Words and Repetitions in Words. CRM Monograph Series. American Mathematical Society, Providence (2008)Google Scholar
  4. 4.
    Berthé, V., de Luca, A., Reutenauer, C.: On an involution of Christoffel words and Sturmian morphisms. Eur. J. Comb. 29, 535–553 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bombieri, E.: Continued fractions and the Markoff tree. Expo. Math. 25, 187–213 (2007)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Borel, J.-P., Laubie, F.: Quelques mots sur la droite projective réelle. Journal de Théorie des Nombres de Bordeaux 5, 23–51 (1993)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Brlek, S., Lachaud, J.O., Provençal, X., Reutenauer, C.: Lyndon + Christoffel = digitally convex. Pattern Recogn. 42, 2239–2246 (2009)CrossRefGoogle Scholar
  8. 8.
    Calkin, N., Wilf, H.S.: Recounting the rationals. Am. Math. Mon. 107, 360–363 (2000)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cassels, J.W.S.: The Markoff chain. Ann. Math. 50, 676–685 (1949)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cassels, J.W.S.: An Introduction to Diophantine Approximation. Cambridge University Press, Cambridge (1957)zbMATHGoogle Scholar
  11. 11.
    Christoffel, E.B.: Observatio arithmetica. Annali di Matematica Pura ed Applicata 6, 145–152 (1875)zbMATHGoogle Scholar
  12. 12.
    Cohn, H.: Approach to Markoff’s minimal forms through modular functions. Ann. Math. 61, 1–12 (1955)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Cohn, H.: Growth types of Fibonacci and Markoff. Fibonacci Quart. 17, 178–183 (1979)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Cusick, T.W., Flahive, M.E.: The Markoff and Lagrange Spectra. American Mathematical Society, Providence (1989)CrossRefGoogle Scholar
  15. 15.
    de Luca, A.: Sturmian words: structure, combinatorics, and their arithmetics. Theor. Comput. Sci. 183, 45–82 (1997)MathSciNetCrossRefGoogle Scholar
  16. 16.
    de Luca, A., de Luca, A.: Pseudopalindrome closure operators in free monoids. Theor. Comput. Sci. 362(1–3), 282–300 (2006)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Dickson L.E.: Studies in the Theory of Numbers. Chelsea, New York (1957) (first edition 1930)Google Scholar
  18. 18.
    Fricke, R.: Ueber die Theorie der automorphen Modulgruppen. Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen 91–101 (1896)Google Scholar
  19. 19.
    Frobenius, G.F.: Über die Markoffschen Zahlen. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin 26, 458–487 (1913)zbMATHGoogle Scholar
  20. 20.
    Graham, R.L., Knuth, D., Patashnik, O.: Concrete Mathematics, 2nd edn. Addison Wesley, Reading (1994)zbMATHGoogle Scholar
  21. 21.
    Hurwitz, A.: Ueber die angenäherte Darstellung der Irrationalzahlen durch rational Brüche. Mathematische Annalen 39, 279–284 (1891)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Justin, J.: Episturmian morphisms and a Galois theorem on continued fractions. Theor. Inform. Appl. 39, 207–215 (2005)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Klette, R., Rosenfeld, A.: Digital straightness – a review. Discret. Appl. Math. 139, 197–230 (2004)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Korkine, A., Zolotareff, G.: Sur les formes quadratiques. Mathematische Annalen 6, 366–389 (1873)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Lothaire, M.: Algebraic Combinatorics on Words. Cambridge University Press, Cambridge (2002)CrossRefGoogle Scholar
  26. 26.
    Markoff, A.A.: Sur les formes quadratiques binaires indéfinies. Mathematische Annalen 15, 381–496 (1879)CrossRefGoogle Scholar
  27. 27.
    Markoff, A.A.: Sur les formes quadratiques binaires indéfinies (second mémoire). Mathematische Annalen 17, 379–399 (1880)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Morse, M., Hedlund, G.A.: Symbolic dynamics II: sturmian trajectories. Am. J. Math. 62, 1–42 (1940)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Perrine, S.: La théorie de Markoff et ses développements. Tessier et Ashpool, Chantilly (2002)Google Scholar
  30. 30.
    Perron, O.: Über die Approximationen irrationaler Zahlen durch rationale II. Sitzungsbereich der Heidelberger Akademie der Wissenschaften 8, 2–12 (1921)zbMATHGoogle Scholar
  31. 31.
    Remak, R.: Über indefinite binäre quadratische Minimalformen. Mathematische Annalen 92, 155–182 (1924)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Reutenauer, C.: Christoffel words and Markoff triples. Integers 9, 327–332 (2009)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Reutenauer, C.: From Christoffel Words to Markoff Numbers. Oxford University Press (2018, to appear)Google Scholar
  34. 34.
    Reutenauer, C., Vuillon, L.: Palindromic closures and Thue-Morse substitution for Markoff numbers. Unif. Distrib. Theory 12, 25–35 (2017)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Series, C.: The geometry of Markoff numbers. Math. Intell. 7, 20–29 (1985)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Smith, H.J.S.: Note on continued fractions. Messenger Math. 6, 1–14 (1876)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Département de mathématiquesUniversité du Québec à MontréalMontréalCanada

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