Revista Matemática Iberoamericana
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Published online: 2002-12-31
Non-rectifiable limit sets of dimension oneChristopher J. Bishop (1) SUNY at Stony Brook, USA
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.
Keywords: Hausdorff dimension, quasi-Fuchsian groups, quasiconformal deformation, critical exponent, convex core
Bishop Christopher: Non-rectifiable limit sets of dimension one. Rev. Mat. Iberoam. 18 (2002), 653-684. doi: 10.4171/RMI/331