Conical limit sets and continued fractions

Authors:
Edward Crane and Ian Short

Journal:
Conform. Geom. Dyn. **11** (2007), 224-249

MSC (2000):
Primary 51B10; Secondary 40A15

DOI:
https://doi.org/10.1090/S1088-4173-07-00169-5

Published electronically:
October 31, 2007

MathSciNet review:
2354097

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Inspired by questions of convergence in continued fraction theory, Erdos, Piranian and Thron studied the possible sets of divergence for arbitrary sequences of Möbius maps acting on the Riemann sphere, . By identifying with the boundary of three-dimensional hyperbolic space, , we show that these sets of divergence are precisely the sets that arise as conical limit sets of subsets of . Using hyperbolic geometry, we give simple geometric proofs of the theorems of Erdos, Piranian and Thron that generalise to arbitrary dimensions. New results are also obtained about the class of conical limit sets; for example, it is closed under locally quasisymmetric homeomorphisms. Applications are given to continued fractions.

**1.**Beat Aebischer, The limiting behaviour of sequences of Möbius transformations,*Math. Zeit.***205**(1990), 49-59 MR**1069484 (91i:51007)****2.**Glen D. Anderson, Mavina K. Vamanamurthy and Matti K. Vuorinen, Conformal invariants, inequalities, and quasiconformal maps, Canadian Mathematical Society Monographs and Advanced Texts,*John Wiley & Sons, Inc., New York*, 1997. MR**1462077 (98h:30033)****3.**Alan F. Beardon, The Geometry of Discrete Groups, Graduate Texts in Mathematics, 91*Springer-Verlag*, 1983. MR**698777 (85d:22026)****4.**Alan F. Beardon, Continued Fractions, Discrete Groups and Complex Dynamics,*Comput. Methods and Funct. Theory***1**(2001), 535-594. MR**1941142 (2003m:30010)****5.**Alan F. Beardon, The pointwise convergence of Möbius maps,*Michigan Math. J.***52**(2004), 483-489. MR**2097393 (2005g:30051)****6.**Alan F. Beardon and Lisa Lorentzen, Continued fractions and restrained sequences of Möbius maps,*Rocky Mountain J. Math.***34**(2004), 441-446. MR**2072789 (2005e:30002)****7.**Riccardo Benedetti and Carlo Petronio, Lectures on hyperbolic geometry, Universitext,*Springer-Verlag*, 1992. MR**1219310 (94e:57015)****8.**V. A. Efremovic and E. S. Tihomirova, Equimorphisms of hyperbolic spaces (Russian),*Izv. Akad. Nauk SSSR Ser. Mat.***28**(1964), 1139-1144. MR**0169121 (29:6374)****9.**Paul Erdos and George Piranian, Sequences of linear fractional transformations,*Michigan Math. J.***6**(1959), 205-209. MR**0109227 (22:114)****10.**Lisa Jacobsen, General convergence of continued fractions,*Trans. Amer. Math. Soc.***294**(1986), 477-485. MR**825716 (87j:40004)****11.**George Piranian and Wolfgang J. Thron, Convergence properties of sequences of linear fractional transformations,*Michigan Math. J.***4**(1957), 129-135. MR**0093578 (20:102)****12.**John G. Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, 149,*Springer-Verlag*, 1994. MR**1299730 (95j:57011)****13.**Pekka Tukia and Jussi Väisälä, Quasiconformal extension from dimension to ,*Ann. of Math*(2)**115**(1982), 331-348. MR**647809 (84i:30030)****14.**Matti Vuorinen, Conformal geometry and quasiregular mappings, Lecture Notes in Math. 1319,*Springer*, 1988. MR**950174 (89k:30021)**

Retrieve articles in *Conformal Geometry and Dynamics of the American Mathematical Society*
with MSC (2000):
51B10,
40A15

Retrieve articles in all journals with MSC (2000): 51B10, 40A15

Additional Information

**Edward Crane**

Affiliation:
Department of Mathematics, University Walk, Bristol, BS8 1TW, United Kingdom

Email:
Edward.Crane@gmail.com

**Ian Short**

Affiliation:
Logic House, National University of Ireland, Maynooth, Maynooth, County Kildare, Ireland

Email:
Ian.Short@nuim.ie

DOI:
https://doi.org/10.1090/S1088-4173-07-00169-5

Keywords:
Conical limit set,
continued fraction,
hyperbolic geometry,
quasiconformal mapping,
Diophantine approximation

Received by editor(s):
January 3, 2007

Published electronically:
October 31, 2007

Additional Notes:
The first author was supported by a junior research fellowship at Merton College, Oxford, and by the University of Bristol.

The second author was supported by Science Foundation Ireland grant 05/RFP/MAT0003.

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.