## A characterization of harmonic measures on laminations by hyperbolic Riemann surfaces.(English)Zbl 1189.37033

The authors study measures associated to compact nonsingular laminations by hyperbolic Riemann surfaces. It builds on a previous article [M. Martínez, Ergodic Theory Dyn. Syst. 26, No. 3, 847–867 (2006; Zbl 1107.37027)] by the second author, and improves on the main results found therein.
Let $${\mathcal L}$$ denote a lamination (or foliated space), which is compact and whose leaves are hyperbolic Riemann surfaces. Each leaf has its Poincaré metric – the metric of constant curvature -1 compatible with the conformal structure. The authors consider two different kinds of measures associated to $${\mathcal L}$$: measures invariant under the heat diffusion along the leaves of $${\mathcal L}$$ which are called harmonic, and measures on the unit tangent bundle $$T^1{\mathcal L}$$ of $${\mathcal L}$$ that are invariant under the laminated geodesic and horocycle flows. Let $$\mu$$ be a probability measure on $${\mathcal L}$$. It is proved that $$\mu$$ is harmonic if and only if there is a measure $$\nu$$ on $$T^1{\mathcal L}$$ which is invariant under both the geodesic and stable horocycle flow and that projects onto $$\mu$$ (under the canonical projection $$T^1{\mathcal L} \to {\mathcal L}$$). Furthermore, such a $$\nu$$ is unique.

### MSC:

 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 58J65 Diffusion processes and stochastic analysis on manifolds 60J65 Brownian motion

Zbl 1107.37027
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### References:

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