**MSC:**- 30F40 Kleinian groups, See also {20H10}
- 30F35 Fuchsian groups and automorphic functions, See also {11Fxx, 20H10, 22E40, 32Gxx, 32Nxx}

$\rho$-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 $\delta_\rho$.