Commensurable continued fractions

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[6]	_______, Cross sections for geodesic flows and $\alpha$-continued fractions, Nonlinearity, 26 (2013), 711-726. doi: 10.1088/0951-7715/26/3/711.
[7]	_______, Natural extensions for piecewise Möbius interval maps,, in preparation., ().
[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.
[9]	K. Dajani and C. Kraaikamp, Ergodic Theory of Numbers, Carus Mathematical Monographs, 29. Mathematical Association of America, Washington, DC, 2002.
[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.
[11]	D. Fried, Symbolic dynamics for triangle groups, Invent. Math., 125 (1996), 487-521. doi: 10.1007/s002220050084.
[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.
[13]	E. Hopf, Fuchsian groups and ergodic theory, Trans. Amer. Math. Soc., 39 (1936), 299-314. doi: 10.1090/S0002-9947-1936-1501848-8.
[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.
[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.
[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.
[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.
[18]	H. Nakada, Continued fractions, geodesic flows and Ford circles, in Algorithms, fractals, and dynamics (Okayama/Kyoto, 1992), Plenum, New York, (1995), 179-191.
[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.
[20]	D. Rosen, A class of continued fractions associated with certain properly discontinuous groups, Duke Math. J., 21 (1954), 549-563.
[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.
[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.
[23]	T. A. Schmidt and M. Sheingorn, Length spectra of the Hecke triangle groups, Math. Z., 220 (1995), 369-397. doi: 10.1007/BF02572621.
[24]	F. Schweiger, Ergodic Theory of Fibred Systems and Metric Number Theory, Oxford: Clarendon Press, 1995.
[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.
[26]	________, The billiard in a regular polygon, Geom. Funct. Anal., 2 (1992), 341-379. doi: 10.1007/BF01896876.

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.
[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.
[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.
[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.
[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.
[6]	_______, Cross sections for geodesic flows and $\alpha$-continued fractions, Nonlinearity, 26 (2013), 711-726. doi: 10.1088/0951-7715/26/3/711.
[7]	_______, Natural extensions for piecewise Möbius interval maps,, in preparation., ().
[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.
[9]	K. Dajani and C. Kraaikamp, Ergodic Theory of Numbers, Carus Mathematical Monographs, 29. Mathematical Association of America, Washington, DC, 2002.
[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.
[11]	D. Fried, Symbolic dynamics for triangle groups, Invent. Math., 125 (1996), 487-521. doi: 10.1007/s002220050084.
[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.
[13]	E. Hopf, Fuchsian groups and ergodic theory, Trans. Amer. Math. Soc., 39 (1936), 299-314. doi: 10.1090/S0002-9947-1936-1501848-8.
[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.
[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.
[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.
[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.
[18]	H. Nakada, Continued fractions, geodesic flows and Ford circles, in Algorithms, fractals, and dynamics (Okayama/Kyoto, 1992), Plenum, New York, (1995), 179-191.
[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.
[20]	D. Rosen, A class of continued fractions associated with certain properly discontinuous groups, Duke Math. J., 21 (1954), 549-563.
[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.
[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.
[23]	T. A. Schmidt and M. Sheingorn, Length spectra of the Hecke triangle groups, Math. Z., 220 (1995), 369-397. doi: 10.1007/BF02572621.
[24]	F. Schweiger, Ergodic Theory of Fibred Systems and Metric Number Theory, Oxford: Clarendon Press, 1995.
[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.
[26]	________, The billiard in a regular polygon, Geom. Funct. Anal., 2 (1992), 341-379. doi: 10.1007/BF01896876.

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