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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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

J. Bourgain and A. Kontorovich, On Zaremba's Conjecture, C. R. Math. Acad. Sci. Paris, 349 (2011), 493-495.  Google Scholar

[4]

P. Flajolet and B. Vallée, On the Gauss-Kuzmin-Wirsing constant, unpublished note, 1995.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

[5]

H. Heilbronn, On the average length of a class of finite continued fractions,, in, (): 87.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

D. Hensley, The number of steps in the Euclidean algorithm, J. Num. Th., 49 (1994), 142-182. doi: 10.1006/jnth.1994.1088.  Google Scholar

[9]

D. Hensley, "Continued Fractions,'' World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. doi: 10.1142/9789812774682.  Google Scholar

[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.  Google Scholar

[11]

F. Schweiger, "Multidimensional Continued Fractions," Oxford Science Publications, Oxford University Press, Oxford, 2000.  Google Scholar

[12]

E. Wirsing, On the theorem of Gauss-Kusmin-Lévy and a Frobenius-type theorem for function spaces,, V. Acta Arith., 24 (): 507.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

J. Bourgain and A. Kontorovich, On Zaremba's Conjecture, C. R. Math. Acad. Sci. Paris, 349 (2011), 493-495.  Google Scholar

[4]

P. Flajolet and B. Vallée, On the Gauss-Kuzmin-Wirsing constant, unpublished note, 1995.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

[5]

H. Heilbronn, On the average length of a class of finite continued fractions,, in, (): 87.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

D. Hensley, The number of steps in the Euclidean algorithm, J. Num. Th., 49 (1994), 142-182. doi: 10.1006/jnth.1994.1088.  Google Scholar

[9]

D. Hensley, "Continued Fractions,'' World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. doi: 10.1142/9789812774682.  Google Scholar

[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.  Google Scholar

[11]

F. Schweiger, "Multidimensional Continued Fractions," Oxford Science Publications, Oxford University Press, Oxford, 2000.  Google Scholar

[12]

E. Wirsing, On the theorem of Gauss-Kusmin-Lévy and a Frobenius-type theorem for function spaces,, V. Acta Arith., 24 (): 507.   Google Scholar

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