Infinite ergodic theory for Kleinian groups

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 (an), which are governed by the exponent of convergence δ and the maximal possible rank kmax of the parabolic elements of the group as follows an    nmax for δ < (kmax + 1)/2 n/ logn for δ = (kmax + 1)/2 n for δ > (kmax + 1)/2. 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 (an), which then allows to determine the index of variation β = min{1, 2δ − kmax}, and to deduce that the ergodic sums Sn(f)/an converge strongly distributional to the MittagLeffler distribution of index β. 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. Prepared using etds.cls [Version: 1999/07/21 v1.0] 2 M. Stadlbauer and B. O. Stratmann Introduction and statement of main results In this paper we study the action of a Kleinian group on its limit set by using methods from non–invertible infinite ergodic theory. Recall that a Kleinian group is a discrete subgroup of the group of orientation preserving isometries of hyperbolic 3–space H = H (for which we shall mainly use the Poincaré ball model equipped with the hyperbolic metric d = dH (see e.g. [9])). Throughout, we exclusively consider essentially free Kleinian groups G, that is we assume that G admits the choice of a Poincaré polyhedron F (see [20]) with finitely many faces such that if two faces s and t of F intersect inside H, then the two associated generators gs and gt of G commute. By Poincaré’s theorem (see [12]), we hence have that an essentially free Kleinian group has no relations other than those which originate from cusps of rank 2. Also, note that groups in this class are in particular geometrically finite. Our first aim is to construct a coding map T associated with G. This construction generalises the well–known Bowen–Series map (see [11, 28]) to Kleinian groups of the second kind, that is to groups G whose limit set L(G) does not coincide with the whole boundary ∂H of hyperbolic space. In particular, T is an endomorphism of the radial limit set Lr(G), which is the intersection of L(G) with the complement of the set of parabolic fixed points of G. In order to obtain a canonical T –invariant measure ν, we then employ the well–known Patterson measure and its associated Liouville–Patterson measure (see [25, 32]). More precisely, by specifying a Poincaré section, we show how to obtain a measure ν̃ which is invariant under the first–return map S. The map S will also be referred to as a section map. It then turns out that T is a factor of S, and we obtain our measure ν by a straight–forward disintegration procedure. The following theorem gives the main results of this paper. In here kmax refers to the maximal possible rank of the parabolic fixed points of G, and δ = δ(G) denotes the exponent of convergence of G, that is the abzissa of convergence of the Poincaré series ∑ g∈G exp(−sd(0, g(0))). Main Theorem. The coding map T : Lr(G) → Lr(G) is a is a topologically mixing Markov map which is conservative and ergodic with respect to ν. If G has no parabolic elements, then ν is finite. If G has parabolic elements, then ν is infinite if and only if δ ≤ (kmax + 1)/2. Moreover, the following holds. In here T̂ refers to the dual of T . 1. If ν is finite then the return sequence (an) of T is given by Birkhoff’s theorem. That is an = n for all n ∈ N, and