Continued fractions, Cantor sets, Hausdorff dimension, and transfer operators and their analytic extension

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July 2012, 32(7): 2417-2436. doi: 10.3934/dcds.2012.32.2417

Continued fractions, Cantor sets, Hausdorff dimension, and transfer operators and their analytic extension

Doug Hensley ^1,

1.	Mathematics Department, Texas A&M University, College Station, TX 77843-3368, United States

Received December 2009 Revised August 2011 Published March 2012

Abstract
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We survey the dynamical systems side of the theory of continued fractions and touch on some of the frontiers of the subject. Ergodic theory plays a role. The work of Baladi and Vallée is discussed. Power series methods that allow for the computation of various numbers such as the Hausdorff dimension of a continued fraction Cantor set, or the Wirsing constant of a particular continued fraction algorithm, to high accuracy, are also discussed.

Keywords: dynamical system, Euclidean algorithm, Hurwitz complex continued fraction., Hausdorff dimension, continued fractions, bounded partial quotients, invariant density.

Mathematics Subject Classification: Primary: 11A55, 11K50, 11K55, 11J70; Secondary: 11K60, 11Y60, 11Y65, 30B1.

Citation: Doug Hensley. Continued fractions, Cantor sets, Hausdorff dimension, and transfer operators and their analytic extension. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2417-2436. doi: 10.3934/dcds.2012.32.2417

References:

[1]	V. Baladi and B. Vallée, Euclidean algorithms are Gaussian, J. Num. Th., 110 (2005), 331-386. doi: 10.1016/j.jnt.2004.08.008.
[2]	W. Bosma, H. Jager and F. Wiedijk, Some metrical observations on the approximation by continued fractions, Nederl. Akad. Wetensch. Indag. Math., 45 (1983), 281-299.
[3]	J. Bourgain and A. Kontorovich, On Zaremba's Conjecture, C. R. Math. Acad. Sci. Paris, 349 (2011), 493-495.
[4]	P. Flajolet and B. Vallée, On the Gauss-Kuzmin-Wirsing constant, unpublished note, 1995.
[5]	H. Heilbronn, On the average length of a class of finite continued fractions, in "1969 Number Theory and Analysis (Papers in Honor of Edmund Landau)," Plenum, New York, 87-96.
[6]	D. Hensley, The distribution of badly approximable rationals and continuants with bounded digits. II, J. Num. Th., 34 (1990), 293-334. doi: 10.1016/0022-314X(90)90139-I.
[7]	D. Hensley, A polynomial time algorithm for the Hausdorff dimension of continued fraction Cantor sets, J. Num. Th., 58 (1996), 9-45. doi: 10.1006/jnth.1996.0058.
[8]	D. Hensley, The number of steps in the Euclidean algorithm, J. Num. Th., 49 (1994), 142-182. doi: 10.1006/jnth.1994.1088.
[9]	D. Hensley, "Continued Fractions,'' World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. doi: 10.1142/9789812774682.
[10]	R. Nair, On metric Diophantine approximation and subsequence ergodic theory, in "New York J. Math.," 3A, (1997/98), Proceedings of the New York Journal of Mathematics Conference, June 9-13, (1997), 117-124.
[11]	F. Schweiger, "Multidimensional Continued Fractions," Oxford Science Publications, Oxford University Press, Oxford, 2000.
[12]	E. Wirsing, On the theorem of Gauss-Kusmin-Lévy and a Frobenius-type theorem for function spaces, V. Acta Arith., 24 (1973/74), 507-528.

show all references

References:

[1]	V. Baladi and B. Vallée, Euclidean algorithms are Gaussian, J. Num. Th., 110 (2005), 331-386. doi: 10.1016/j.jnt.2004.08.008.
[2]	W. Bosma, H. Jager and F. Wiedijk, Some metrical observations on the approximation by continued fractions, Nederl. Akad. Wetensch. Indag. Math., 45 (1983), 281-299.
[3]	J. Bourgain and A. Kontorovich, On Zaremba's Conjecture, C. R. Math. Acad. Sci. Paris, 349 (2011), 493-495.
[4]	P. Flajolet and B. Vallée, On the Gauss-Kuzmin-Wirsing constant, unpublished note, 1995.
[5]	H. Heilbronn, On the average length of a class of finite continued fractions, in "1969 Number Theory and Analysis (Papers in Honor of Edmund Landau)," Plenum, New York, 87-96.
[6]	D. Hensley, The distribution of badly approximable rationals and continuants with bounded digits. II, J. Num. Th., 34 (1990), 293-334. doi: 10.1016/0022-314X(90)90139-I.
[7]	D. Hensley, A polynomial time algorithm for the Hausdorff dimension of continued fraction Cantor sets, J. Num. Th., 58 (1996), 9-45. doi: 10.1006/jnth.1996.0058.
[8]	D. Hensley, The number of steps in the Euclidean algorithm, J. Num. Th., 49 (1994), 142-182. doi: 10.1006/jnth.1994.1088.
[9]	D. Hensley, "Continued Fractions,'' World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. doi: 10.1142/9789812774682.
[10]	R. Nair, On metric Diophantine approximation and subsequence ergodic theory, in "New York J. Math.," 3A, (1997/98), Proceedings of the New York Journal of Mathematics Conference, June 9-13, (1997), 117-124.
[11]	F. Schweiger, "Multidimensional Continued Fractions," Oxford Science Publications, Oxford University Press, Oxford, 2000.
[12]	E. Wirsing, On the theorem of Gauss-Kusmin-Lévy and a Frobenius-type theorem for function spaces, V. Acta Arith., 24 (1973/74), 507-528.

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