Arkiv för Matematik

  • Ark. Mat.
  • Volume 35, Number 2 (1997), 201-224.

Wiggly sets and limit sets

Christopher J. Bishop and Peter W. Jones

Full-text: Open access

Abstract

We show that a compact, connected set which has uniform oscillations at all points and at all scales has dimension strictly larger than 1. We also show that limit sets of certain Kleinian groups have this property. More generally, we show that if G is a non-elementary, analytically finite Kleinian group, and its limit set Λ(G) is connected, then Λ(G) is either a circle or has dimension strictly bigger than 1.

Note

The first author is partially supported by NSF Grant DMS 95-00577 and an Alfred P. Sloan research fellowship. The second author is partially supported by NSF grant DMS-94-23746.

Article information

Source
Ark. Mat. Volume 35, Number 2 (1997), 201-224.

Dates
Received: 29 January 1996
Revised: 10 February 1997
First available in Project Euclid: 31 January 2017

Permanent link to this document
http://projecteuclid.org/euclid.afm/1485898549

Digital Object Identifier
doi:10.1007/BF02559967

Mathematical Reviews number (MathSciNet)
MR1478778

Zentralblatt MATH identifier
0939.30031

Rights
1997 © Institut Mittag-Leffler

Citation

Bishop, Christopher J.; Jones, Peter W. Wiggly sets and limit sets. Ark. Mat. 35 (1997), no. 2, 201--224. doi:10.1007/BF02559967. http://projecteuclid.org/euclid.afm/1485898549.


Export citation

References

  • Abikoff, W. and Maskit, B., Geometric decompositions of Kleinian groups, Amer. J. Math. 99 (1977), 687–697.
  • Ahlfors, L. V., Finitely generated Kleinian groups, Amer. J. Math. 86 (1964), 413–429.
  • Ahlfors, L. V., Lectures on Quasiconformal Mappings, Math. Studies 10, Van Nostrand, Toronto-New York-London, 1966.
  • Ahlfors, L. V., Conformal Invariants: Topics in Geometric Function Theory, McGraw-Hill, New York-Düsseldorf-Johannesburg, 1973.
  • Astala, K. and Zinsmeister, M., Mostow rigidity and Fuchsian groups, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), 301–306.
  • Bers, L., Inequalities for finitely generated Kleinian groups, J. Analyse Math. 18 (1967), 23–41.
  • Bishop, C. J. and Jones, P. W., Harmonic measure and arclength, Ann. of Math. 132 (1990), 511–547.
  • Bishop, C. J. and Jones, P. W., Harmonic measure, L2 estimates and the Schwarzian derivative, J. Analyse Math. 62 (1994), 77–113.
  • Bishop, C. J. and Jones, P. W., Hausdorff dimension and Kleinian groups, Acta Math. 179 (1997), 1–39.
  • Bishop, C. J., Jones, P. W., Pemantle, R. and Peres, Y., The dimension of the Brownian frontier is greater than 1, J. Funct. Anal. 143 (1997), 309–336.
  • Bowen, R., Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 11–25.
  • Braam, P., A Kaluza-Klein approach to hyperbolic three-manifolds, Enseign. Math. 34 (1988), 275–311.
  • Bullett, S. and Mantica, G., Group theory of hyperbolic circle packings, Nonlinearity 5 (1992), 1085–1109.
  • Canary, R. D., The Poincaré metric and a conformal version of a theorem of Thurston, Duke Math. J. 64 (1991), 349–359.
  • Canary, R. D. and Taylor, E., Kleinian groups with small limit sets, Duke Math. J. 73 (1994), 371–381.
  • Coifman, R., Jones, P. W. and Semmes, S., Two elementary proofs of the L2 boundedness of Cauchy integrals on Lipschitz graphs, J. Amer. Math. Soc. 2 (1989), 553–564.
  • Duren, P., Univalent Functions, Springer-Verlag, Berlin-Heidelberg, 1983.
  • Furusawa, H., The exponent of convergence of Poincaré series of combination groups, Tôhoku Math. J. 43 (1991), 1–7.
  • Garnett, J. B., Bounded Analytic Functions, Academic Press, Orlando, Fla., 1981.
  • Jerison, D. S. and Kenig, C. E., Hardy spaces, A and singular integrals on chord-arc domains, Math. Scand. 50 (1982), 221–248.
  • Jones, P. W., Rectifiable sets and the traveling salesman problem, Invent. Math. 102 (1990), 1–15.
  • Keen, L., Maskit, B. and Series, C., Geometric finiteness and uniqueness for Kleinian groups with circle packing limit sets, J. Reine Angew. Math. 436 (1993), 209–219.
  • Larman, D. H., On the Besicowitch dimension of the residual set of aubitrary packed disks in the plane, J. London Math. Soc., 42 (1967), 292–302.
  • Maskit, B., Kleinian Groups, Springer-Verlag, Berlin-Heidelberg, 1988.
  • McShanh, G., Parker, J. R. and Redfern, I., Drawing limit sets of Kleinian groups using finite state automata, Experiment. Math. 3 (1994), 153–170.
  • Nicholls, P. J., The Ergodic Theory of Discrette Groups, London Math. Soc. Lecture Note Ser., 143, Cambridge Univ. Press, Cambridge, 1989.
  • Okikiolu, K., Characterizations of subsets of nectifiable curves in RnJ. London Math. Soc. 46 (1992), 336–348.
  • Parker, J. R., Kleinian circle packings, Topology 34 (1995), 489–496.
  • Pommeronke, C. Polymonphic finctions for groups of divergence type, Math. Ann. 258 (1982), 353–366.
  • Pommeronke, C., On uniformly perfect sets and Fuchsian groups, Analysis 4 (1984), 299–321.
  • Rohde, S., On conformal welding and quasicricles, Michigan. Math. J. 38 (1991). 111–116.
  • Stein, E., Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N. J., 1970.
  • Sulliwan, D., The density at infinity of a discrete group of hyperbolic motions, Inst. Hantes Études Sci. Publ. Math. 50 (1979), 172–202.
  • Sullivan, D., Discrete conformal groups and measureable dynamics, Bull. Amer. Math. Soc. 6 (1982), 57–73.
  • Tomaschitz, R., Quantum chaos on hyperbolic manifolds: a new approach to cosmology, Compler Systems 6 (1992), 137–161.
  • Väisälä, J., Bilipschitz and quasisymmetric extension properties, Ann. Acad. Sci. Fem. Ser. A I Math. 11 (1986), 239–274.
  • Väisälä, J., Vuorinen, M., and Wallin, H., Thick sets and quasisymmetric maps, Nagoya Math. J. 135 (1994), 121–148.
  • Wheden, R. and Zygmund, A., Measure and Indegral, Marcel Dekker, New York, 1977.