Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Continued fractions with $ SL(2, Z)$-branches: combinatorics and entropy

Authors: Carlo Carminati, Stefano Isola and Giulio Tiozzo
Journal: Trans. Amer. Math. Soc. 370 (2018), 4927-4973
MSC (2010): Primary 11A55, 11K50, 37A10
Published electronically: February 21, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the dynamics of a family $ K_\alpha $ of discontinuous interval maps whose (infinitely many) branches are Möbius transformations in $ SL(2, \mathbb{Z})$ and which arise as the critical-line case of the family of $ (a, b)$-continued fractions.

We provide an explicit construction of the bifurcation locus $ \mathcal {E}_{KU}$ for this family, showing it is parametrized by Farey words and it has Hausdorff dimension zero. As a consequence, we prove that the metric entropy of $ K_\alpha $ is analytic outside the bifurcation set but not differentiable at points of $ \mathcal {E}_{KU}$ and that the entropy is monotone as a function of the parameter.

Finally, we prove that the bifurcation set is combinatorially isomorphic to the main cardioid in the Mandelbrot set, providing one more entry to the dictionary developed by the authors between continued fractions and complex dynamics.

References [Enhancements On Off] (What's this?)

  • [1] Pierre Arnoux and Thomas A. Schmidt, Cross sections for geodesic flows and $ \alpha$-continued fractions, Nonlinearity 26 (2013), no. 3, 711-726. MR 3018939,
  • [2] Jean Berstel, Sturmian and episturmian words (a survey of some recent results), Algebraic informatics, Lecture Notes in Comput. Sci., vol. 4728, Springer, Berlin, 2007, pp. 23-47. MR 2681739,
  • [3] Jean Berstel, Aaron Lauve, Christophe Reutenauer, and Franco V. Saliola, Combinatorics on words, Christoffel words and repetitions in words, CRM Monograph Series, vol. 27, American Mathematical Society, Providence, RI, 2009. MR 2464862
  • [4] Gamaliel Blé, External arguments and invariant measures for the quadratic family, Discrete Contin. Dyn. Syst. 11 (2004), no. 2-3, 241-260. MR 2083418,
  • [5] Enrico Bombieri, Continued fractions and the Markoff tree, Expo. Math. 25 (2007), no. 3, 187-213. MR 2345177,
  • [6] Claudio Bonanno, Carlo Carminati, Stefano Isola, and Giulio Tiozzo, Dynamics of continued fractions and kneading sequences of unimodal maps, Discrete Contin. Dyn. Syst. 33 (2013), no. 4, 1313-1332. MR 2995847,
  • [7] S. Brlek, J.-O. Lachaud, X. Provençal, and C. Reutenauer, Lyndon + Christoffel = digitally convex, Pattern Recognition 42 (2009), 2239-2246.
  • [8] Shaun Bullett and Pierrette Sentenac, Ordered orbits of the shift, square roots, and the devil's staircase, Math. Proc. Cambridge Philos. Soc. 115 (1994), no. 3, 451-481 (English, with English and French summaries). MR 1269932,
  • [9] Carlo Carminati, Stefano Marmi, Alessandro Profeti, and Giulio Tiozzo, The entropy of $ \alpha$-continued fractions: numerical results, Nonlinearity 23 (2010), no. 10, 2429-2456. MR 2672682,
  • [10] Carlo Carminati and Giulio Tiozzo, A canonical thickening of $ \mathbb{Q}$ and the entropy of $ \alpha$-continued fraction transformations, Ergodic Theory Dynam. Systems 32 (2012), no. 4, 1249-1269. MR 2955313,
  • [11] C. Carminati and G. Tiozzo, The bifurcation locus for the set of bounded type numbers, arXiv:1109.0516 [math.DS].
  • [12] Carlo Carminati and Giulio Tiozzo, Tuning and plateaux for the entropy of $ \alpha$-continued fractions, Nonlinearity 26 (2013), no. 4, 1049-1070. MR 3040595,
  • [13] Robert L. Devaney, The Mandelbrot set, the Farey tree, and the Fibonacci sequence, Amer. Math. Monthly 106 (1999), no. 4, 289-302. MR 1682393,
  • [14] Kenneth Falconer, Fractal geometry, Mathematical foundations and applications, John Wiley & Sons, Ltd., Chichester, 1990. MR 1102677
  • [15] N. Pytheas Fogg, Substitutions in dynamics, arithmetics and combinatorics, edited by V. Berthé, S. Ferenczi, C. Mauduit and A.Siegel, Lecture Notes in Mathematics, vol. 1794, Springer-Verlag, Berlin, 2002. MR 1970385
  • [16] Jane Gilman and Linda Keen, Enumerating palindromes and primitives in rank two free groups, J. Algebra 332 (2011), 1-13. MR 2774675,
  • [17] Lisa R. Goldberg, Fixed points of polynomial maps. I. Rotation subsets of the circles, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 6, 679-685. MR 1198093
  • [18] Marston Morse and Gustav A. Hedlund, Symbolic dynamics II. Sturmian trajectories, Amer. J. Math. 62 (1940), 1-42. MR 0000745,
  • [19] J. H. Hubbard, Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz, Topological methods in modern mathematics (Stony Brook, NY, 1991) Publish or Perish, Houston, TX, 1993, pp. 467-511. MR 1215974
  • [20] John H. Hubbard and Colin T. Sparrow, The classification of topologically expansive Lorenz maps, Comm. Pure Appl. Math. 43 (1990), no. 4, 431-443. MR 1047331,
  • [21] A. Hurwitz, Über eine besondere Art der Kettenbruch-Entwicklung reeller Grössen, Acta Math. 12 (1889), no. 1, 367-405 (German). MR 1554778,
  • [22] Marius Iosifescu and Cor Kraaikamp, Metrical theory of continued fractions, Mathematics and its Applications, vol. 547, Kluwer Academic Publishers, Dordrecht, 2002. MR 1960327
  • [23] Svetlana Katok, Coding of closed geodesics after Gauss and Morse, Geom. Dedicata 63 (1996), no. 2, 123-145. MR 1413625,
  • [24] Svetlana Katok and Ilie Ugarcovici, Structure of attractors for $ (a,b)$-continued fraction transformations, J. Mod. Dyn. 4 (2010), no. 4, 637-691. MR 2753948
  • [25] Svetlana Katok and Ilie Ugarcovici, Applications of $ (a,b)$-continued fraction transformations, Ergodic Theory Dynam. Systems 32 (2012), no. 2, 755-777. MR 2901369,
  • [26] Linda Keen and Caroline Series, Pleating coordinates for the Maskit embedding of the Teichmüller space of punctured tori, Topology 32 (1993), no. 4, 719-749. MR 1241870,
  • [27] Cor Kraaikamp, A new class of continued fraction expansions, Acta Arith. 57 (1991), no. 1, 1-39. MR 1093246
  • [28] Cor Kraaikamp, Thomas A. Schmidt, and Wolfgang Steiner, Natural extensions and entropy of $ \alpha$-continued fractions, Nonlinearity 25 (2012), no. 8, 2207-2243. MR 2946184,
  • [29] Rafael Labarca and Carlos Gustavo Moreira, Essential dynamics for Lorenz maps on the real line and the lexicographical world, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), no. 5, 683-694 (English, with English and French summaries). MR 2259612,
  • [30] M. Lothaire, Combinatorics on words, with a foreword by Roger Lyndon and a preface by Dominique Perrin, corrected reprint of the 1983 original, with a new preface by Perrin, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1997. MR 1475463
  • [31] Laura Luzzi and Stefano Marmi, On the entropy of Japanese continued fractions, Discrete Contin. Dyn. Syst. 20 (2008), no. 3, 673-711. MR 2373210
  • [32] S. Mantaci, A. Restivo, and M. Sciortino, Burrows-Wheeler transform and Sturmian words, Inform. Process. Lett. 86 (2003), no. 5, 241-246. MR 1976388,
  • [33] John Milnor, Periodic orbits, externals rays and the Mandelbrot set: an expository account, Géométrie complexe et systèmes dynamiques (Orsay, 1995), Astérisque 261 (2000), xiii, 277-333 (English, with English and French summaries). MR 1755445
  • [34] Hitoshi Nakada, Metrical theory for a class of continued fraction transformations and their natural extensions, Tokyo J. Math. 4 (1981), no. 2, 399-426. MR 646050,
  • [35] Hitoshi Nakada and Rie Natsui, The non-monotonicity of the entropy of $ \alpha$-continued fraction transformations, Nonlinearity 21 (2008), no. 6, 1207-1225. MR 2422375,
  • [36] Giulio Tiozzo, The entropy of Nakada's $ \alpha$-continued fractions: analytical results, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13 (2014), no. 4, 1009-1037. MR 3362117
  • [37] Giulio Tiozzo, Entropy, dimension and combinatorial moduli for one-dimensional dynamical systems, ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.)-Harvard University, 2013. MR 3167282
  • [38] D. B. Zagier, Zetafunktionen und quadratische Körper, Eine Einführung in die höhere Zahlentheorie. [An introduction to higher number theory], Hochschultext. [University Text], Springer-Verlag, Berlin-New York, 1981 (German). MR 631688

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11A55, 11K50, 37A10

Retrieve articles in all journals with MSC (2010): 11A55, 11K50, 37A10

Additional Information

Carlo Carminati
Affiliation: Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy

Stefano Isola
Affiliation: Scuola di Scienze e Tecnologie, Università di Camerino, via Madonna delle Carceri, 62032 Camerino, Italy

Giulio Tiozzo
Affiliation: Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario M5S 2E4, Canada

Received by editor(s): February 21, 2014
Received by editor(s) in revised form: October 14, 2016
Published electronically: February 21, 2018
Additional Notes: The authors acknowledge the support of the CRM “Ennio de Giorgi” of Pisa and the program “Dynamical Numbers” at the Max Planck Institute of Bonn.
The first author was partially supported by the GNAMPA group of the Istituto Nazionale di Alta Matematica (INdAM) and the MIUR project “Teorie geometriche e analitiche dei sistemi Hamiltoniani in dimensioni finite e infinite” (PRIN 2010JJ4KPA_008).
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society