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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
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show all references

References:
[1]

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]

Bull. Amer. Math. Soc. (N.S.), 25 (1991), 229-334. doi: 10.1090/S0273-0979-1991-16076-3.  Google Scholar

[3]

(French) [Coding of the geodesic flow on the modular surface], Enseign. Math. (2), 40 (1994), 29-48.  Google Scholar

[4]

(French) [Continued fractions on Veech's surfaces], J. Anal. Math., 81 (2000), 35-64. doi: 10.1007/BF02788985.  Google Scholar

[5]

J. Mod. Dyn., 3 (2009), 611-629. doi: 10.3934/jmd.2009.3.611.  Google Scholar

[6]

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]

Trans. Amer. Math. Soc., 352 (2000), 1277-1298. doi: 10.1090/S0002-9947-99-02442-3.  Google Scholar

[9]

Carus Mathematical Monographs, 29. Mathematical Association of America, Washington, DC, 2002.  Google Scholar

[10]

J. Eur. Math. Soc. (JEMS), 11 (2009), 1259-1283. doi: 10.4171/JEMS/181.  Google Scholar

[11]

Invent. Math., 125 (1996), 487-521. doi: 10.1007/s002220050084.  Google Scholar

[12]

Canad. Math. Bull., 39 (1996), 186-198. doi: 10.4153/CMB-1996-023-8.  Google Scholar

[13]

Trans. Amer. Math. Soc., 39 (1936), 299-314. doi: 10.1090/S0002-9947-1936-1501848-8.  Google Scholar

[14]

Duke Math. J., 123 (2004), 49-69. doi: 10.1215/S0012-7094-04-12312-8.  Google Scholar

[15]

Nonlinearity, 25 (2012), 2207-2243. doi: 10.1088/0951-7715/25/8/2207.  Google Scholar

[16]

in Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989), Oxford Sci. Publ., Oxford Univ. Press, New York, (1991), 71-91.  Google Scholar

[17]

J. Mod. Dyn., 2 (2008), 581-627. doi: 10.3934/jmd.2008.2.581.  Google Scholar

[18]

in Algorithms, fractals, and dynamics (Okayama/Kyoto, 1992), Plenum, New York, (1995), 179-191.  Google Scholar

[19]

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]

Duke Math. J., 21 (1954), 549-563.  Google Scholar

[21]

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]

J. London Math. Soc. (2), 31 (1985), 69-80. doi: 10.1112/jlms/s2-31.1.69.  Google Scholar

[23]

Math. Z., 220 (1995), 369-397. doi: 10.1007/BF02572621.  Google Scholar

[24]

Oxford: Clarendon Press, 1995.  Google Scholar

[25]

Inv. Math., 97 (1989), 553-583. doi: 10.1007/BF01388890.  Google Scholar

[26]

Geom. Funct. Anal., 2 (1992), 341-379. doi: 10.1007/BF01896876.  Google Scholar

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