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Wiggly sets and limit sets

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Arkiv för Matematik

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 ifG 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.

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References

  1. Abikoff, W. andMaskit, B., Geometric decompositions of Kleinian groups,Amer. J. Math. 99 (1977), 687–697.

    Article  MATH  MathSciNet  Google Scholar 

  2. Ahlfors, L. V., Finitely generated Kleinian groups,Amer. J. Math. 86 (1964), 413–429.

    Article  MathSciNet  Google Scholar 

  3. Ahlfors, L. V.,Lectures on Quasiconformal Mappings, Math. Studies10, Van Nostrand, Toronto-New York-London, 1966.

    MATH  Google Scholar 

  4. Ahlfors, L. V.,Conformal Invariants: Topics in Geometric Function Theory, McGraw-Hill, New York-Düsseldorf-Johannesburg, 1973.

    MATH  Google Scholar 

  5. Astala, K. andZinsmeister, M., Mostow rigidity and Fuchsian groups,C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), 301–306.

    MATH  MathSciNet  Google Scholar 

  6. Bers, L., Inequalities for finitely generated Kleinian groups,J. Analyse Math. 18 (1967), 23–41.

    Article  MATH  MathSciNet  Google Scholar 

  7. Bishop, C. J. andJones, P. W., Harmonic measure and arclength,Ann. of Math. 132 (1990), 511–547.

    Article  MathSciNet  Google Scholar 

  8. Bishop, C. J. andJones, P. W., Harmonic measure,L 2 estimates and the Schwarzian derivative,J. Analyse Math. 62 (1994), 77–113.

    Article  MATH  MathSciNet  Google Scholar 

  9. Bishop, C. J. andJones, P. W., Hausdorff dimension and Kleinian groups,Acta Math. 179 (1997), 1–39.

    Article  MATH  MathSciNet  Google Scholar 

  10. Bishop, C. J., Jones, P. W., Pemantle, R. andPeres, Y., The dimension of the Brownian frontier is greater than 1,J. Funct. Anal. 143 (1997), 309–336.

    Article  MATH  MathSciNet  Google Scholar 

  11. Bowen, R., Hausdorff dimension of quasicircles,Inst. Hautes Études Sci. Publ. Math. 50 (1979), 11–25.

    Article  MATH  MathSciNet  Google Scholar 

  12. Braam, P., A Kaluza-Klein approach to hyperbolic three-manifolds,Enseign. Math. 34 (1988), 275–311.

    MATH  MathSciNet  Google Scholar 

  13. Bullett, S. andMantica, G., Group theory of hyperbolic circle packings,Nonlinearity 5 (1992), 1085–1109.

    Article  MATH  MathSciNet  Google Scholar 

  14. Canary, R. D., The Poincaré metric and a conformal version of a theorem of Thurston,Duke Math. J. 64 (1991), 349–359.

    Article  MATH  MathSciNet  Google Scholar 

  15. Canary, R. D. andTaylor, E., Kleinian groups with small limit sets,Duke Math. J. 73 (1994), 371–381.

    Article  MATH  MathSciNet  Google Scholar 

  16. Coifman, R., Jones, P. W. andSemmes, S., Two elementary proofs of theL 2 boundedness of Cauchy integrals on Lipschitz graphs,J. Amer. Math. Soc. 2 (1989), 553–564.

    Article  MATH  MathSciNet  Google Scholar 

  17. Duren, P.,Univalent Functions, Springer-Verlag, Berlin-Heidelberg, 1983.

    MATH  Google Scholar 

  18. Furusawa, H., The exponent of convergence of Poincaré series of combination groups,Tôhoku Math. J. 43 (1991), 1–7.

    Article  MATH  MathSciNet  Google Scholar 

  19. Garnett, J. B.,Bounded Analytic Functions, Academic Press, Orlando, Fla., 1981.

    MATH  Google Scholar 

  20. Jerison, D. S. andKenig, C. E., Hardy spaces,A ∞ and singular integrals on chord-arc domains,Math. Scand. 50 (1982), 221–248.

    MATH  MathSciNet  Google Scholar 

  21. Jones, P. W., Rectifiable sets and the traveling salesman problem,Invent. Math. 102 (1990), 1–15.

    Article  MATH  MathSciNet  Google Scholar 

  22. Keen, L., Maskit, B. andSeries, C., Geometric finiteness and uniqueness for Kleinian groups with circle packing limit sets,J. Reine Angew. Math. 436 (1993), 209–219.

    MATH  MathSciNet  Google Scholar 

  23. 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.

    Article  MATH  MathSciNet  Google Scholar 

  24. Maskit, B.,Kleinian Groups, Springer-Verlag, Berlin-Heidelberg, 1988.

    MATH  Google Scholar 

  25. McShanh, G., Parker, J. R. andRedfern, I., Drawing limit sets of Kleinian groups using finite state automata,Experiment. Math. 3 (1994), 153–170.

    MathSciNet  Google Scholar 

  26. Nicholls, P. J.,The Ergodic Theory of Discrette Groups, London Math. Soc. Lecture Note Ser.,143, Cambridge Univ. Press, Cambridge, 1989.

    Google Scholar 

  27. Okikiolu, K., Characterizations of subsets of nectifiable curves inR n J. London Math. Soc. 46 (1992), 336–348.

    Article  MATH  MathSciNet  Google Scholar 

  28. Parker, J. R., Kleinian circle packings,Topology 34 (1995), 489–496.

    Article  MATH  MathSciNet  Google Scholar 

  29. Pommeronke, C. Polymonphic finctions for groups of divergence type,Math. Ann. 258 (1982), 353–366.

    Article  Google Scholar 

  30. Pommeronke, C., On uniformly perfect sets and Fuchsian groups,Analysis 4 (1984), 299–321.

    MathSciNet  Google Scholar 

  31. Rohde, S., On conformal welding and quasicricles,Michigan. Math. J. 38 (1991). 111–116.

    Article  MATH  MathSciNet  Google Scholar 

  32. Stein, E.,Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N. J., 1970.

    MATH  Google Scholar 

  33. Sulliwan, D., The density at infinity of a discrete group of hyperbolic motions,Inst. Hantes Études Sci. Publ. Math. 50 (1979), 172–202.

    Google Scholar 

  34. Sullivan, D., Discrete conformal groups and measureable dynamics,Bull. Amer. Math. Soc. 6 (1982), 57–73.

    Article  MATH  MathSciNet  Google Scholar 

  35. Tomaschitz, R., Quantum chaos on hyperbolic manifolds: a new approach to cosmology,Compler Systems 6 (1992), 137–161.

    MATH  MathSciNet  Google Scholar 

  36. Väisälä, J., Bilipschitz and quasisymmetric extension properties,Ann. Acad. Sci. Fem. Ser. A I Math. 11 (1986), 239–274.

    MATH  Google Scholar 

  37. Väisälä, J., Vuorinen, M., andWallin, H., Thick sets and quasisymmetric maps,Nagoya Math. J. 135 (1994), 121–148.

    MATH  MathSciNet  Google Scholar 

  38. Wheden, R. andZygmund, A.,Measure and Indegral, Marcel Dekker, New York, 1977.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematics Department, State University of New York at Stony Brook, II794-3651, Stony Brook, NY, USA

    Christopher J. Bishop

  2. Mathematics Department, Yale University, 06520, New Haven, CT, USA

    Peter W. Jones

Authors
  1. Christopher J. Bishop
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  2. Peter W. Jones
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Additional information

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.

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Bishop, C.J., Jones, P.W. Wiggly sets and limit sets. Ark. Mat. 35, 201–224 (1997). https://doi.org/10.1007/BF02559967

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  • DOI: https://doi.org/10.1007/BF02559967

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