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November  2014, 34(11): 4389-4418. doi: 10.3934/dcds.2014.34.4389

Commensurable continued fractions

Pierre Arnoux 1, and  Thomas A. Schmidt 2,

1. 

Institut de Mathématiques de Luminy (UMR6206 CNRS), 163 Avenue de Luminy, case 907, 13288 Marseille cedex 09, France
2. 

Department of Mathematics, Oregon State University, Corvallis, OR 97331, United States

Received  September 2013 Revised  February 2014 Published  May 2014

We compare two families of continued fractions algorithms, the symmetrized Rosen algorithm and the Veech algorithm. Each of these algorithms expands real numbers in terms of certain algebraic integers. We give explicit models of the natural extension of the maps associated with these algorithms; prove that these natural extensions are in fact conjugate to the first return map of the geodesic flow on a related surface; and, deduce that, up to a conjugacy, almost every real number has an infinite number of common approximants for both algorithms.
Keywords: Moebius transformations, Fuchsian groups., geodesic flow, Continued fractions, natural extension.
Mathematics Subject Classification: Primary: 37E05, 11K50; Secondary: 37D40, 30B7.
Citation: Pierre Arnoux, Thomas A. Schmidt. Commensurable continued fractions. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4389-4418. doi: 10.3934/dcds.2014.34.4389
References:
[1]

R. Adler, Continued fractions and Bernoulli trials, in Ergodic Theory (A Seminar), (eds. J. Moser, E. Phillips and S. Varadhan), Courant Inst. of Math. Sci. (Lect. Notes 110), 1975, New York.  Google Scholar

[2]

R. Adler and L. Flatto, Geodesic flows, interval maps, and symbolic dynamics, Bull. Amer. Math. Soc. (N.S.), 25 (1991), 229-334. doi: 10.1090/S0273-0979-1991-16076-3.  Google Scholar

[3]

P. Arnoux, Le codage du flot géodésique sur la surface modulaire, (French) [Coding of the geodesic flow on the modular surface], Enseign. Math. (2), 40 (1994), 29-48.  Google Scholar

[4]

P. Arnoux and P. Hubert, Fractions continues sur les surfaces de Veech, (French) [Continued fractions on Veech's surfaces], J. Anal. Math., 81 (2000), 35-64. doi: 10.1007/BF02788985.  Google Scholar

[5]

P. Arnoux and T. A. Schmidt, Veech surfaces with non-periodic directions in the trace field, J. Mod. Dyn., 3 (2009), 611-629. doi: 10.3934/jmd.2009.3.611.  Google Scholar

[6]

_______, Cross sections for geodesic flows and $\alpha$-continued fractions, Nonlinearity, 26 (2013), 711-726. doi: 10.1088/0951-7715/26/3/711.  Google Scholar

[7]

_______, Natural extensions for piecewise Möbius interval maps,, in preparation., ().   Google Scholar

[8]

R. Burton, C. Kraaikamp and T. A. Schmidt, Natural extensions for the Rosen fractions, Trans. Amer. Math. Soc., 352 (2000), 1277-1298. doi: 10.1090/S0002-9947-99-02442-3.  Google Scholar

[9]

K. Dajani and C. Kraaikamp, Ergodic Theory of Numbers, Carus Mathematical Monographs, 29. Mathematical Association of America, Washington, DC, 2002.  Google Scholar

[10]

K. Dajani, C. Kraaikamp and W. Steiner, Metrical theory for $\alpha$-Rosen fractions, J. Eur. Math. Soc. (JEMS), 11 (2009), 1259-1283. doi: 10.4171/JEMS/181.  Google Scholar

[11]

D. Fried, Symbolic dynamics for triangle groups, Invent. Math., 125 (1996), 487-521. doi: 10.1007/s002220050084.  Google Scholar

[12]

K. Gröchenig and A. Haas, Backward continued fractions and their invariant measures, Canad. Math. Bull., 39 (1996), 186-198. doi: 10.4153/CMB-1996-023-8.  Google Scholar

[13]

E. Hopf, Fuchsian groups and ergodic theory, Trans. Amer. Math. Soc., 39 (1936), 299-314. doi: 10.1090/S0002-9947-1936-1501848-8.  Google Scholar

[14]

P. Hubert and T. A. Schmidt, Infinitely generated Veech groups, Duke Math. J., 123 (2004), 49-69. doi: 10.1215/S0012-7094-04-12312-8.  Google Scholar

[15]

C. Kraaikamp, T. A. Schmidt and W. Steiner, Natural extensions and entropy of $\alpha$-continued fractions, Nonlinearity, 25 (2012), 2207-2243. doi: 10.1088/0951-7715/25/8/2207.  Google Scholar

[16]

A. Manning, Dynamics of geodesic and horocycle flows on surfaces of constant negative curvature, in Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989), Oxford Sci. Publ., Oxford Univ. Press, New York, (1991), 71-91.  Google Scholar

[17]

D. Mayer and F. Strömberg, Symbolic dynamics for the geodesic flow on Hecke surfaces, J. Mod. Dyn., 2 (2008), 581-627. doi: 10.3934/jmd.2008.2.581.  Google Scholar

[18]

H. Nakada, Continued fractions, geodesic flows and Ford circles, in Algorithms, fractals, and dynamics (Okayama/Kyoto, 1992), Plenum, New York, (1995), 179-191.  Google Scholar

[19]

V. A. Rohlin, Exact endomorphisms of Lebesgue spaces (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961), 499-530. Translation in (MR0160698) Amer. Math. Soc. Transl. Series 2, 39 (1964), 1-36.  Google Scholar

[20]

D. Rosen, A class of continued fractions associated with certain properly discontinuous groups, Duke Math. J., 21 (1954), 549-563.  Google Scholar

[21]

T. A. Schmidt, Remarks on the Rosen $\lambda$- continued fractions in Number theory with an emphasis on the Markoff spectrum, (eds. A. Pollington and W. Moran), Dekker, New York, 147 (1993), 227-238.  Google Scholar

[22]

C. Series, The modular surface and continued fractions, J. London Math. Soc. (2), 31 (1985), 69-80. doi: 10.1112/jlms/s2-31.1.69.  Google Scholar

[23]

T. A. Schmidt and M. Sheingorn, Length spectra of the Hecke triangle groups, Math. Z., 220 (1995), 369-397. doi: 10.1007/BF02572621.  Google Scholar

[24]

F. Schweiger, Ergodic Theory of Fibred Systems and Metric Number Theory, Oxford: Clarendon Press, 1995.  Google Scholar

[25]

W. A. Veech, Teichmüller curves in modular space, Eisenstein series, and an application to triangular billiards, Inv. Math., 97 (1989), 553-583. doi: 10.1007/BF01388890.  Google Scholar

[26]

________, The billiard in a regular polygon, Geom. Funct. Anal., 2 (1992), 341-379. doi: 10.1007/BF01896876.  Google Scholar

show all references

References:
[1]

R. Adler, Continued fractions and Bernoulli trials, in Ergodic Theory (A Seminar), (eds. J. Moser, E. Phillips and S. Varadhan), Courant Inst. of Math. Sci. (Lect. Notes 110), 1975, New York.  Google Scholar

[2]

R. Adler and L. Flatto, Geodesic flows, interval maps, and symbolic dynamics, Bull. Amer. Math. Soc. (N.S.), 25 (1991), 229-334. doi: 10.1090/S0273-0979-1991-16076-3.  Google Scholar

[3]

P. Arnoux, Le codage du flot géodésique sur la surface modulaire, (French) [Coding of the geodesic flow on the modular surface], Enseign. Math. (2), 40 (1994), 29-48.  Google Scholar

[4]

P. Arnoux and P. Hubert, Fractions continues sur les surfaces de Veech, (French) [Continued fractions on Veech's surfaces], J. Anal. Math., 81 (2000), 35-64. doi: 10.1007/BF02788985.  Google Scholar

[5]

P. Arnoux and T. A. Schmidt, Veech surfaces with non-periodic directions in the trace field, J. Mod. Dyn., 3 (2009), 611-629. doi: 10.3934/jmd.2009.3.611.  Google Scholar

[6]

_______, Cross sections for geodesic flows and $\alpha$-continued fractions, Nonlinearity, 26 (2013), 711-726. doi: 10.1088/0951-7715/26/3/711.  Google Scholar

[7]

_______, Natural extensions for piecewise Möbius interval maps,, in preparation., ().   Google Scholar

[8]

R. Burton, C. Kraaikamp and T. A. Schmidt, Natural extensions for the Rosen fractions, Trans. Amer. Math. Soc., 352 (2000), 1277-1298. doi: 10.1090/S0002-9947-99-02442-3.  Google Scholar

[9]

K. Dajani and C. Kraaikamp, Ergodic Theory of Numbers, Carus Mathematical Monographs, 29. Mathematical Association of America, Washington, DC, 2002.  Google Scholar

[10]

K. Dajani, C. Kraaikamp and W. Steiner, Metrical theory for $\alpha$-Rosen fractions, J. Eur. Math. Soc. (JEMS), 11 (2009), 1259-1283. doi: 10.4171/JEMS/181.  Google Scholar

[11]

D. Fried, Symbolic dynamics for triangle groups, Invent. Math., 125 (1996), 487-521. doi: 10.1007/s002220050084.  Google Scholar

[12]

K. Gröchenig and A. Haas, Backward continued fractions and their invariant measures, Canad. Math. Bull., 39 (1996), 186-198. doi: 10.4153/CMB-1996-023-8.  Google Scholar

[13]

E. Hopf, Fuchsian groups and ergodic theory, Trans. Amer. Math. Soc., 39 (1936), 299-314. doi: 10.1090/S0002-9947-1936-1501848-8.  Google Scholar

[14]

P. Hubert and T. A. Schmidt, Infinitely generated Veech groups, Duke Math. J., 123 (2004), 49-69. doi: 10.1215/S0012-7094-04-12312-8.  Google Scholar

[15]

C. Kraaikamp, T. A. Schmidt and W. Steiner, Natural extensions and entropy of $\alpha$-continued fractions, Nonlinearity, 25 (2012), 2207-2243. doi: 10.1088/0951-7715/25/8/2207.  Google Scholar

[16]

A. Manning, Dynamics of geodesic and horocycle flows on surfaces of constant negative curvature, in Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989), Oxford Sci. Publ., Oxford Univ. Press, New York, (1991), 71-91.  Google Scholar

[17]

D. Mayer and F. Strömberg, Symbolic dynamics for the geodesic flow on Hecke surfaces, J. Mod. Dyn., 2 (2008), 581-627. doi: 10.3934/jmd.2008.2.581.  Google Scholar

[18]

H. Nakada, Continued fractions, geodesic flows and Ford circles, in Algorithms, fractals, and dynamics (Okayama/Kyoto, 1992), Plenum, New York, (1995), 179-191.  Google Scholar

[19]

V. A. Rohlin, Exact endomorphisms of Lebesgue spaces (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961), 499-530. Translation in (MR0160698) Amer. Math. Soc. Transl. Series 2, 39 (1964), 1-36.  Google Scholar

[20]

D. Rosen, A class of continued fractions associated with certain properly discontinuous groups, Duke Math. J., 21 (1954), 549-563.  Google Scholar

[21]

T. A. Schmidt, Remarks on the Rosen $\lambda$- continued fractions in Number theory with an emphasis on the Markoff spectrum, (eds. A. Pollington and W. Moran), Dekker, New York, 147 (1993), 227-238.  Google Scholar

[22]

C. Series, The modular surface and continued fractions, J. London Math. Soc. (2), 31 (1985), 69-80. doi: 10.1112/jlms/s2-31.1.69.  Google Scholar

[23]

T. A. Schmidt and M. Sheingorn, Length spectra of the Hecke triangle groups, Math. Z., 220 (1995), 369-397. doi: 10.1007/BF02572621.  Google Scholar

[24]

F. Schweiger, Ergodic Theory of Fibred Systems and Metric Number Theory, Oxford: Clarendon Press, 1995.  Google Scholar

[25]

W. A. Veech, Teichmüller curves in modular space, Eisenstein series, and an application to triangular billiards, Inv. Math., 97 (1989), 553-583. doi: 10.1007/BF01388890.  Google Scholar

[26]

________, The billiard in a regular polygon, Geom. Funct. Anal., 2 (1992), 341-379. doi: 10.1007/BF01896876.  Google Scholar

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