Conformal measures associated to ends of hyperbolic n-manifolds

Anderson, James W., Falk, Kurt and Tukia, Pekka (2007) Conformal measures associated to ends of hyperbolic n-manifolds. Quarterly Journal of Mathematics, 58 (1), 1-15. (doi:10.1093/qmath/hal019).

Record type: Article

Abstract

Let G be a non-elementary Kleinian group acting on the closed n-dimensional unit ball and assume that its Poincare series converges at the exponent a. Let M be the G-quotient of the open unit ball. We consider certain families E={e1, ..., ep} of open subsets of M such that the complement of the union of e1, ..., ep in M is compact. The sets e1, ..., ep are the ends of M and E is a complete collection of ends for M. We associate to each end e in E an a-conformal measure such that the measures corresponding to different ends are mutually singular if non-trivial. Additionally, each a-conformal measure for G on its limit set can be written as a sum of such conformal measures associated to ends e in E. In dimension 3, our results overlap with some results of Bishop and Jones.

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