Fractal dimensions for Jarník limit sets of geometrically finite Kleinian groups; the semi-classical approach

Abstract

We introduce and study the Jarnik limit set ℐσ of a geometrically finite Kleinian group with parabolic elements. The set ℐσ is the dynamical equivalent of the classical set of well approximable limit points. By generalizing the method of Jarnik in the theory of Diophantine approximations, we estimate the dimension of ℐσ with respect to the Patterson measure. In the case in which the exponent of convergence of the group does not exceed the maximal rank of the parabolic fixed points, and hence in particular for all finitely generated Fuchsian groups, it is shown that this leads to a complete description of ℐσ in terms of Hausdorff dimension. For the remaining case, we derive some estimates for the Hausdorff dimension and the packing dimension of ℐσ.