## Conical limit sets and continued fractions

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- by Edward Crane and Ian Short PDF
- Conform. Geom. Dyn.
**11**(2007), 224-249 Request permission

## Abstract:

Inspired by questions of convergence in continued fraction theory, Erdős, Piranian and Thron studied the possible sets of divergence for arbitrary sequences of Möbius maps acting on the Riemann sphere, $S^2$. By identifying $S^2$ with the boundary of three-dimensional hyperbolic space, $H^3$, we show that these sets of divergence are precisely the sets that arise as conical limit sets of subsets of $H^3$. Using hyperbolic geometry, we give simple geometric proofs of the theorems of Erdős, 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.## 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 - Glen D. Anderson, Mavina K. Vamanamurthy, and Matti K. Vuorinen,
*Conformal invariants, inequalities, and quasiconformal maps*, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1997. With 1 IBM-PC floppy disk (3.5 inch; HD); A Wiley-Interscience Publication. MR**1462077** - 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 - A. F. Beardon,
*The pointwise convergence of Möbius maps*, Michigan Math. J.**52**(2004), no. 3, 483–489. MR**2097393**, DOI 10.1307/mmj/1100623408 - A. F. Beardon and L. Lorentzen,
*Continued fractions and restrained sequences of Möbius maps*, Rocky Mountain J. Math.**34**(2004), no. 2, 441–466. MR**2072789**, DOI 10.1216/rmjm/1181069862 - Riccardo Benedetti and Carlo Petronio,
*Lectures on hyperbolic geometry*, Universitext, Springer-Verlag, Berlin, 1992. MR**1219310**, DOI 10.1007/978-3-642-58158-8 - V. A. Efremovič and E. S. Tihomirova,
*Equimorphisms of hyperbolic spaces*, Izv. Akad. Nauk SSSR Ser. Mat.**28**(1964), 1139–1144 (Russian). MR**0169121** - Paul Erdős and George Piranian,
*Sequences of linear fractional transformations*, Michigan Math. J.**6**(1959), 205–209. MR**109227** - 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 - G. Piranian and W. J. Thron,
*Convergence properties of sequences of linear fractional transformations*, Michigan Math. J.**4**(1957), 129–135. MR**93578**, DOI 10.1307/mmj/1028989001 - 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 - P. Tukia and J. Väisälä,
*Quasiconformal extension from dimension $n$ to $n+1$*, Ann. of Math. (2)**115**(1982), no. 2, 331–348. MR**647809**, DOI 10.2307/1971394 - Matti Vuorinen,
*Conformal geometry and quasiregular mappings*, Lecture Notes in Mathematics, vol. 1319, Springer-Verlag, Berlin, 1988. MR**950174**, DOI 10.1007/BFb0077904

## 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
- MR Author ID: 791601
- ORCID: 0000-0002-7360-4089
- Email: Ian.Short@nuim.ie
- 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. - © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2354097