Weak singularity spectra of the Patterson measure for geometrically finite Kleinian groups with parabolic elements
Let \(G\) be a geometrically finite Kleinian group with parabolic elements and \(L(G)\) its limit set. Further let denote \(B(\xi,r)\) the Euclidean ball of radius \(r\) centered at \(\xi\) and \(\mu\) the Patterson measure supported on the limit set \(L(G)\). Let us put for \(\vartheta>0\) \[ I^\vartheta (\mu):=\left \{\xi \in L(G); \liminf_{r\to 0}{\log\mu \bigl(B(\xi,r) \bigr)\over\log r}\leq \vartheta \right\}. \tag{1} \] By turning the inequality in (1), the set \(I_\vartheta (\mu)\) is obtained and changing \(\underline {\text{lim sup}}\) instead \(\underline {\text{lim inf}}\) we obtain further two sets \(S^\vartheta (\mu)\) and \(S_{\vartheta} (\mu)\). In the paper the Hausdorff dimension of these four sets is determined.
scientific article; zbMATH DE number 1548326