Harmonicity of Quasiconformal Measures and Poisson Boundaries of Hyperbolic Spaces
By Chris Connell, Roman Muchnik. 2004
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Abstract.We consider a group Γ of isometries acting on a proper (not necessarily geodesic) δ -hyperbolic space X. For any continuous α-quasiconformal measure ν on ∂X assigning full measure to Λr, the radial limit set of Γ, we produce a (nontrivial) measure μ on Γ for which ν is stationary. This means that the limit set together with ν forms a μ-boundary and ν is harmonic with respect to the random walk induced by μ. As a basic example, take $$X = {\mathbb{H}}^n$$ and Γ to be any geometrically finite Kleinian group with ν a Patterson-Sullivan measure for Γ. In the case when X is a CAT(−1) space and Γ is discrete with quasiconvex action, we show that (Λr, ν) is the 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 L1 or L∞ norm, for the space of uniformly positive lower-semicontinuous functions on a general metric measure space.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/7b11b6b8ef8190806344df53e9e5a58a8a08246e
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- Harmonic measures versus quasiconformal measures for hyperbolic groups (2008, background): “In a different spirit, it is proved in [14] that any Patterson-Sullivan measure can be realized as the harmonic measure of some random walk.”