CONFORMAL MEASURES ASSOCIATED TO ENDS OF HYPERBOLIC n-MANIFOLDS

Abstract

Let Γ be a non-elementary Kleinian group acting on the closed n-dimensional unit ball and assume that its Poincaré series converges at the exponent α. Let MΓ be the Γ-quotient of the open unit ball. We consider certain families ℰ = {E1, …, Ep} of open subsets of MΓ such that is compact. The sets Ei are the ends of MΓ and ℰ is a complete collection of ends for MΓ. We associate to each end E ∈ ℰ an α-conformal measure such that the measures corresponding to different ends are mutually singular if non-trivial. Additionally, each α-conformal measure for Γ on the limit set Λ(Γ) of Γ can be written as a sum of such conformal measures associated to ends E ∈ ℰ. In dimension 3, our results overlap with some results of Bishop and Jones (The law of the iterated logarithm for Kleinian groups, Cont. Math.211 (1997), 17–50.).