Open Access
October, 2002 Non-rectifiable limit sets of dimension one
Christopher J. Bishop
Rev. Mat. Iberoamericana 18(3): 653-684 (October, 2002).


We construct 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. It is known that no divergence type group has such a deformation. The main tools in this construction are (1) a characterization of tangent points in terms of Peter Jones' $\beta$'s, (2) a result of Stephen Semmes that gives a Carleson type condition on a Beltrami coefficient which implies rectifiability and (3) a construction of quasiconformal deformations of a surface which shrink a given geodesic and whose dilatations satisfy an exponential decay estimate away from the geodesic.


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Christopher J. Bishop. "Non-rectifiable limit sets of dimension one." Rev. Mat. Iberoamericana 18 (3) 653 - 684, October, 2002.


Published: October, 2002
First available in Project Euclid: 28 April 2003

zbMATH: 1064.30046
MathSciNet: MR1954867

Primary: 30F60

Keywords: convex core , Critical exponent , Hausdorff dimension , quasiconformal deformation , Quasi-Fuchsian groups

Rights: Copyright © 2002 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.18 • No. 3 • October, 2002
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