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Non-rectifiable limit sets of dimension one.

Christopher J. Bishop — 2002

Revista Matemática Iberoamericana

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)...

Quasiconformal mappings of Y-pieces.

Christopher J. Bishop — 2002

Revista Matemática Iberoamericana

In this paper we construct quasiconformal mappings between Y-pieces so that the corresponding Beltrami coefficient has exponential decay away from the boundary. These maps are used in a companion paper to construct quasiFuchsian groups whose limit sets are non-rectifiable curves of dimension 1.

An indestructible Blaschke product in the little Bloch space.

Christopher J. Bishop — 1993

Publicacions Matemàtiques

The little Bloch space, B, is the space of all holomorphic functions f on the unit disk such that lim lf'(z)l (1- lzl) = 0. Finite Blaschke products are clearly in B, but examples of infinite products in B are more difficult to obtain (there are now several constructions due to Sarason, Stephenson and the author, among others). Stephenson has asked whether B contains an infinite, indestructible Blaschke product, i.e., a Blaschke product B so that (B(z) - a)/(1 - âB(z)), is also a Blaschke product...

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