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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Abikoff, W. andMaskit, 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. Studies10, 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. andZinsmeister, 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. andJones, P. W., Harmonic measure and arclength,Ann. of Math. 132 (1990), 511–547.
Bishop, C. J. andJones, P. W., Harmonic measure,L 2 estimates and the Schwarzian derivative,J. Analyse Math. 62 (1994), 77–113.
Bishop, C. J. andJones, P. W., Hausdorff dimension and Kleinian groups,Acta Math. 179 (1997), 1–39.
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.
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. andMantica, 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. andTaylor, E., Kleinian groups with small limit sets,Duke Math. J. 73 (1994), 371–381.
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.
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. andKenig, 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. andSeries, 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. andRedfern, 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 inR n J. 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., andWallin, H., Thick sets and quasisymmetric maps,Nagoya Math. J. 135 (1994), 121–148.
Wheden, R. andZygmund, A.,Measure and Indegral, Marcel Dekker, New York, 1977.
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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