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Uniqueness of conformal densities and the semiflow of Weyl chambers. (English. Abridged French version) Zbl 0912.53035

Summary: Let \(X=G/K\) be a symmetric space of noncompact type and \(\Gamma\) a discrete “generic” subgroup of \(G\) with critical exponent \(\delta(\Gamma)\). We show that, if \(\Gamma\) is of divergence type, then there is a unique \(\Gamma\)-invariant conformal density of dimension \(\delta(\Gamma)\) (hence a Patterson-Sullivan density) on the set of regular elements of the geometric boundary of \(X\). This problem is directly related to the recurrence of the semiflow of Weyl chambers on \(X/\Gamma\).

MSC:

53C35 Differential geometry of symmetric spaces
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
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