Infinite ergodic theory for Kleinian groups

Abstract

In this paper we use infinite ergodic theory to study limit sets of essentially free Kleinian groups which may have parabolic elements of arbitrary rank. By adapting a method of Adler, we construct a section map S for the geodesic flow on the associated hyperbolic manifold. We then show that this map has the Markov property and that it is conservative and ergodic with respect to the invariant measure induced by the Liouville–Patterson measure. Furthermore, we obtain that S is rationally ergodic with respect to different types of return sequences (a_n), which are governed by the exponent of convergence $\delta$ and the maximal possible rank k_max of the parabolic elements of the group as follows \[a_n\asymp\begin{cases}n^{2\delta-k_{\max}}&\text{for }\delta<(k_{\max}+1)/2,\\n/\!\log n&\text{for }\delta=(k_{\max}+1)/2,\\n&\text{for }\delta>(k_{\max}+1)/2.\end{cases}\] Subsequently, we give a discussion of an associated canonical map T, which is an analogue of the Bowen–Series map in the Fuchsian case. We show that T is pointwise dual ergodic with respect to these return sequences (a_n), which then allows us to determine the index of variation $\beta=\min\{1,2\delta-k_{\max}\}$, and to deduce that the ergodic sums S_n(f)/a_n converge strongly distributional to the Mittag–Leffler distribution of index $\beta$. We then give applications to number theory and to the statistics of cuspidal windings. Also, as a corollary we obtain a special case of Sullivan's result that the geodesic flow on a geometrically finite hyperbolic manifold is ergodic with respect to the Liouville–Patterson measure.