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Papers by Chris Connell
GAFA Geometric And Functional Analysis, 2007
We consider a group Γ of isometries acting on a (not necessarily geodesic) δ-hyperbolic space and... more We consider a group Γ of isometries acting on a (not necessarily geodesic) δ-hyperbolic space and possessing a radial limit set of full measure within its limit set. For any continuous α-quasiconformal measure ν supported on the limit set, we produce a stationary measure µ on Γ. Moreover the limit set together with ν forms a µ-boundary and ν is harmonic with respect to the random walk induced by µ. In the case when X is a CAT(−κ) space and Γ acts cocompactly, for instance, we show that µ has finite first moment. This implies that (∂X, ν) is the unique Poisson boundary for µ. In the course of the proofs, we establish sufficient conditions for a set of continuous functions to form a positive basis, either in the L 1 or L ∞ norm, for the space of uniformly positive lower-semicontinuous functions on a metric measure space.
GAFA Geometric And Functional Analysis, 2007
We consider a group Γ of isometries acting on a (not necessarily geodesic) δ-hyperbolic space and... more We consider a group Γ of isometries acting on a (not necessarily geodesic) δ-hyperbolic space and possessing a radial limit set of full measure within its limit set. For any continuous α-quasiconformal measure ν supported on the limit set, we produce a stationary measure µ on Γ. Moreover the limit set together with ν forms a µ-boundary and ν is harmonic with respect to the random walk induced by µ. In the case when X is a CAT(−κ) space and Γ acts cocompactly, for instance, we show that µ has finite first moment. This implies that (∂X, ν) is the unique Poisson boundary for µ. In the course of the proofs, we establish sufficient conditions for a set of continuous functions to form a positive basis, either in the L 1 or L ∞ norm, for the space of uniformly positive lower-semicontinuous functions on a metric measure space.