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On the rational ergodicity and ergodic properties of geodesic flows on hyperbolic manifolds. (Sur l’ergodicité rationnelle et les propriétés ergodiques du flot géodésique dans les variétés hyperboliques.) (French) Zbl 0968.37012

Consider a hyperbolic (or more generally CAT(\(-1\))) space \(\mathbb{H}\), equipped with some Kleinian subgroup \(\Gamma\) and a \(\Gamma\)-invariant conformal density \((\mu_x)\) of dimension \(\delta=\dim_H (\Lambda(\Gamma))\). Let \(m\) denote the associated Patterson-Sullivan-Bowen-Margulis measure on the unitary tangent bundle \(T^1\mathbb{H}/ \Gamma\), and \(g\) denote the geodesic flow on \(T^1\mathbb{H}/ \Gamma\).
The author first establishes a complete general version of the Hopf-Tsuji-Sullivan theorem, characterizing the divergence of the Poincaré series at \(\delta\) by the conservativity or the ergodicity of \((q,m)\), or in terms of the support of \(\mu\) and the conical limit set \(\Lambda(\Gamma)\).
Considering then this divergence type case, several finer results of equirepartition are established.
In particular, the so-called rational ergodicity is proved for \((g,m)\), where the asymptotic type \(m(A)^{-2} \int^T_0 m(A \cap g^{-t}A) dt\) is given by the partial sums (till distance \(T)\) of the Poincaré series at \(\delta\). And a weak mixing property of multiple order is established for \((g,m)\). The method of this beautiful article is based on careful estimates of return functions of the following form: \[ \int_0^{T_1} \cdots \int_0^{T_p} \sum_{\gamma_1, \dots,\gamma_p \in\Gamma} m[V_0\cap g^{-t_1} \gamma_1V_1 \cap\cdots \cap g^{-t_1- \cdots-t_p} \gamma_pV_p] dt_1\cdots dt_p, \] in terms of partial multiple sums of Poincaré series at \(\delta\), weighted by \((\mu_x)\)-measures of some shadows.

MSC:

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
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