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

1. 

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

Received  December 2009 Revised  August 2011 Published  March 2012

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P. Flajolet and B. Vallée, On the Gauss-Kuzmin-Wirsing constant, unpublished note, 1995.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