On the dimension distortions of quasi-symmetric homeomorphisms

In this paper, we first generalize a result of Bishop and Steger [Representation theoretic rigidity in PSL(2, R). Acta Math., 170, (1993), 121-149] by proving 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 for any $E\subset \mathbb{R}$, we have max(dim$E$, dim$h(\mathbb{R}\setminus E))=1$. Furthermore, we showed that Bishop and steger's result does not hold for the covering groups of all '$d$-dimensional jungle gym' (d is any positive integer) which generalizes G\"onye's results [ Differentiability of quasi-conformal maps on the jungle gym. Trans. Amer. Math. Soc. Vol 359 (2007), 9-32] where the author discussed the case of '$1$-dimensional jungle gym'.