K. Falk, B.O. Stratmann
Constructing Restricted Patterson Measures for Geometrically Infinite Kleinian Groups
Preprint series: Mathematica Gottingensis
30F40 Kleinian groups, See also {20H10}
30F35 Fuchsian groups and automorphic functions, See also {11Fxx, 20H10, 22E40, 32Gxx, 32Nxx}
Abstract: In this paper we study exhaustions, referred to as
$\rho$-restrictions, of arbitrary non-elementary Kleinian groups with at most
finitely many bounded parabolic elements. Special emphasis is put on the
geometrically infinite case, where we obtain that
the limit set of each of these Kleinian groups contains an infinite family
of closed subsets, referred to as $\rho$-restricted limit sets, such that there
is a Poincar\'e series and hence an exponent of convergence $\delta_\rho$,
canonically associated to every element in this family. Generalizing concepts
which are well-known in the geometrically finite case, we then introduce the
notion of $\rho$-restricted Patterson measure, and show that these measures are
non-atomic, $\delta_\rho$-harmonic, $\delta_\rho$-subconformal on special sets
and $\delta_\rho$-conformal on very special sets. Furthermore, we obtain
the results that each $\rho$-restriction of our Kleinian group is of
$\delta_\rho$-divergence type and that the Hausdorff dimension of the
$\rho$-restricted limit set is equal to $\delta_\rho$.
Keywords: Conformal Measures; Kleinian Groups; Fractal Geometry