×

Non-rectifiable limit sets of dimension one. (English) Zbl 1064.30046

The author considers and constructs quasiconformal deformations of convergence type Fuchsian groups such that the resulting limit set is a Jordan curve of Hausdorff dimension 1, but having tangents almost nowhere. He answers the question, asked by M. Zinsmeisters and K. Astala [C. R. Acad. Sci., Paris, Sér. I 311, 301–306 (1990; Zbl 0709.30039)], that whether there are quasi-Fuchsian groups whose limit set are not locally rectifiable, but still have dimension 1, by showing that there are many such groups.

MSC:

30F60 Teichmüller theory for Riemann surfaces

Citations:

Zbl 0709.30039
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] Astala, K. and Zinsmeister, M.: Mostow rigidity and Fuchsian groups C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), 301-306. · Zbl 0709.30039
[2] Bishop, C. J.: Quasiconformal mappings of Y -pieces. Rev. Mat. Iberoamericana 18 (2002), 627-652. · Zbl 1064.30045
[3] Bishop, C. J.: Divergence groups have the Bowen property. Ann. of Math. 154 (2001), 205-217. · Zbl 0999.37030
[4] Bishop, C. J. and Jones, P.W.: Compact deformations of Fuchsian groups. To appear in J. Anal. Math. · Zbl 1032.30028
[5] Bishop, C. J. and Jones, P.W.: Harmonic measure, L2 estimates and the Schwarzian derivative. J. Anal. Math. 62 (1994), 77-113. · Zbl 0801.30024
[6] Bowen, R.: Hausdorff dimension of quasicircles. Inst. Hautes Études Sci. Publ. Math. 50 (1979), 11-25. · Zbl 0439.30032
[7] Garnett, J.B.: Bounded analytic functions. Academic Press, 1981. · Zbl 0469.30024
[8] Jerison, D.S. and Kenig, C.E.: Hardy spaces, A\infty and singular integrals on chord-arc domains. Math. Scand. 50 (1982), 221-248. · Zbl 0509.30025
[9] Jones, P.W.: Rectifiable sets and travelling salesman problem. Invent. Math. 102 (1990), 1-15. · Zbl 0731.30018
[10] Keen, L.: Collars on Riemann surfaces. In Discontinuous groups and Rie- mann surfaces, Ann. of Math. Stud. 79, Princeton Univ. Press, Princeton, N.J., 1974, 263-268. · Zbl 0304.30014
[11] Matelski, J. P.: A compactness theorem for Fuchsian groups of the sec- ond kind. Duke Math. J. 43 (1976), no. 4, 829-840. · Zbl 0341.30020
[12] Pommerenke, Ch.: Boundary behaviour of conformal maps. Grundlehren Math. Wiss. 299, Springer-Verlag, Berlin, 1992. · Zbl 0762.30001
[13] Semmes, S.: Quasiconformal mappings and chord-arc curves. Trans. Amer. Math. Soc. 306 (1988), 233-263. · Zbl 0653.30008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.