Non-rectifiable limit sets of dimension one

  • Christopher J. Bishop

    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' '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.

Cite this article

Christopher J. Bishop, Non-rectifiable limit sets of dimension one. Rev. Mat. Iberoam. 18 (2002), no. 3, pp. 653–684

DOI 10.4171/RMI/331