Differentiability of quasi-conformal maps on the jungle gym
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Abstract:
We obtain a result on the quasi-conformal self-maps of jungle gyms, a divergence-type group. If the dilatation is compactly supported, then the induced map on the boundary of the covering disc $\mathbb {D}$ is differentiable with non-zero derivative on a set of Hausdorff dimension $1$. As one of the corollaries, we show that there are quasi-symmetric homeomorphisms over divergence-type groups such that for all sets $E$ the Hausdorff dimension of $E$ and $f(E^c)$ cannot both be less than $1$. This shows an important difference between finitely generated and divergence-type groups.References
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Additional Information
- Zsuzsanna Gönye
- Affiliation: Department of Mathematics, Polytechnic University, Brooklyn, New York 11201
- Email: zgonye@poly.edu
- Received by editor(s): September 9, 2004
- Published electronically: August 15, 2006
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 19-32
- MSC (2000): Primary 30C62, 28A78; Secondary 30F35
- DOI: https://doi.org/10.1090/S0002-9947-06-04198-5
- MathSciNet review: 2247880