Hausdorff dimension of sets of divergence arising from continued fractions

Short, Ian

doi:10.1090/S0002-9939-2011-11032-3

Hausdorff dimension of sets of divergence arising from continued fractions
HTML articles powered by AMS MathViewer

by Ian Short PDF

Proc. Amer. Math. Soc. 140 (2012), 1371-1385 Request permission

Abstract:

A complex continued fraction can be represented by a sequence of Möbius transformations in such a way that the continued fraction converges if and only if the sequence converges at the origin. The set of divergence of the sequence of Möbius transformations is equivalent to the conical limit set from Kleinian group theory, and it is closely related to the Julia set from complex dynamics. We determine the Hausdorff dimensions of sets of divergence for sequences of Möbius transformations corresponding to certain important classes of continued fractions.

References

Beat Aebischer, The limiting behavior of sequences of Möbius transformations, Math. Z. 205 (1990), no. 1, 49–59. MR 1069484, DOI 10.1007/BF02571224
Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777, DOI 10.1007/978-1-4612-1146-4
Alan F. Beardon, Continued fractions, discrete groups and complex dynamics, Comput. Methods Funct. Theory 1 (2001), no. 2, [On table of contents: 2002], 535–594. MR 1941142, DOI 10.1007/BF03321006
Alan F. Beardon and Ian Short, The Seidel, Stern, Stolz and Van Vleck theorems on continued fractions, Bull. Lond. Math. Soc. 42 (2010), no. 3, 457–466. MR 2651941, DOI 10.1112/blms/bdq006
Edward Crane and Ian Short, Conical limit sets and continued fractions, Conform. Geom. Dyn. 11 (2007), 224–249. MR 2354097, DOI 10.1090/S1088-4173-07-00169-5
Paul Erdős and George Piranian, Sequences of linear fractional transformations, Michigan Math. J. 6 (1959), 205–209. MR 109227
Kenneth Falconer, Fractal geometry, 2nd ed., John Wiley & Sons, Inc., Hoboken, NJ, 2003. Mathematical foundations and applications. MR 2118797, DOI 10.1002/0470013850
W. K. Hayman and P. B. Kennedy, Subharmonic functions. Vol. I, London Mathematical Society Monographs, No. 9, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. MR 0460672
Lisa Jacobsen, General convergence of continued fractions, Trans. Amer. Math. Soc. 294 (1986), no. 2, 477–485. MR 825716, DOI 10.1090/S0002-9947-1986-0825716-1
William B. Jones and Wolfgang J. Thron, Continued fractions, Encyclopedia of Mathematics and its Applications, vol. 11, Addison-Wesley Publishing Co., Reading, Mass., 1980. Analytic theory and applications; With a foreword by Felix E. Browder; With an introduction by Peter Henrici. MR 595864
Lisa Lorentzen and Haakon Waadeland, Continued fractions. Vol. 1, 2nd ed., Atlantis Studies in Mathematics for Engineering and Science, vol. 1, Atlantis Press, Paris; World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. Convergence theory. MR 2433845, DOI 10.2991/978-94-91216-37-4
R. Daniel Mauldin and Mariusz Urbański, Conformal iterated function systems with applications to the geometry of continued fractions, Trans. Amer. Math. Soc. 351 (1999), no. 12, 4995–5025. MR 1487636, DOI 10.1090/S0002-9947-99-02268-0
Peter J. Nicholls, The ergodic theory of discrete groups, London Mathematical Society Lecture Note Series, vol. 143, Cambridge University Press, Cambridge, 1989. MR 1041575, DOI 10.1017/CBO9780511600678
G. Piranian and W. J. Thron, Convergence properties of sequences of linear fractional transformations, Michigan Math. J. 4 (1957), 129–135. MR 93578
John G. Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, vol. 149, Springer-Verlag, New York, 1994. MR 1299730, DOI 10.1007/978-1-4757-4013-4
C. A. Rogers, Hausdorff measures, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1998. Reprint of the 1970 original; With a foreword by K. J. Falconer. MR 1692618
Bao-Wei Wang and Jun Wu, Hausdorff dimension of certain sets arising in continued fraction expansions, Adv. Math. 218 (2008), no. 5, 1319–1339. MR 2419924, DOI 10.1016/j.aim.2008.03.006