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

Doug Hensley

doi:10.3934/dcds.2012.32.2417

Article Contents

2012, Volume 32, Issue 7: 2417-2436. Doi: 10.3934/dcds.2012.32.2417

This issue Previous Article Examples of coarse expanding conformal maps Next Article On the asymptotic behaviour of the Lebesgue measure of sum-level sets for continued fractions

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

Received: December 2009

Revised: August 2011

Published: July 2012

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 11A55, 11K50, 11K55, 11J70; Secondary: 11K60, 11Y60, 11Y65, 30B10.

Citation:

Related Papers

Cited by

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[4]	P. Flajolet and B. Vallée, On the Gauss-Kuzmin-Wirsing constant, unpublished note, 1995.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.