On the Hausdorff dimension distortions of quasi-symmetric homeomorphisms

Abstract

In this paper, we first prove that for a Fuchsian group \(G\) of divergence type and non-lattice, if \(h\) is a quasi-symmetric homeomorphism of the real axis \(\mathbb{R}\) corresponding to a quasi-conformal compact deformation of \(G\), then \(h\) is not strongly singular for divergence groups. This generalizes a result of Bishop and Steger (1993). Furthermore, we show that Bishop and Steger's result does not hold for the covering groups of all \(d\)-dimensional 'Jungle Gyms' (\(d\) is any positive integer) which generalizes Gönye's results (2007) where the author discussed the case of 1-dimensional 'Jungle Gym'.