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.


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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