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**Ergodicity and equidistribution in negative curvature.
(Ergodicité et équidistribution en courbure négative.)**
*(French)*
Zbl 1056.37034

Mém. Soc. Math. Fr., Nouv. Sér. 95, 96 p. (2003).

In this long article the author proves with elementary methods a number of ergodicity results for \(\operatorname{cat}(-1)\) spaces \(X\), geodesic metric spaces with curvature in the sense of Alexandrov \(\leq - 1\). Examples include simply connected Riemannian manifolds with sectional curvature \(\leq -1\), but also trees and other nonmanifold examples. Some of the results are previously known, especially in the case of Riemannian manifolds, but not in the full generality of this article. A fundamental object for such spaces is the boundary at infinity \(\partial X\), which consists of asymptote classes of geodesics of \(X\). Typically one considers a discrete group \(\Gamma\) acting on \(X\) and by extension to \(\partial X\). Such a group defines a limit set \(\Lambda(\Gamma) \subset \partial X\), which is the set of accumulation points in \(\partial X\) of the \(\Gamma\) orbits in \(X\). The set \(\Lambda(\Gamma)\) equals \(\partial X\) if \(X/\Gamma\) is compact, but in general is a proper closed subset of \(\partial X\). The space \(SX\) of parametrized geodesics in \(X\) agrees with the unit tangent bundle in the Riemannian case, and it may be identified naturally with \(\partial X\times \partial X\times \mathbb{R}\). Relative to this identification the geodesic flow \(\{g^{t} \}\) acts on \(SX\) by \(g^{t}(\zeta, \eta, s) = (\zeta, \eta, s+t)\). The space of horospheres \(\mathfrak{H}\) may be identified with \(\partial X\times \mathbb{R}\). To pursue the ergodic properties of \(\Gamma\) acting on \(\partial X\), \(SX\) or \(\mathfrak{H}\) one uses the Patterson-Sullivan conformal densities \(\{\mu_{x} : x \in X \}\) on \(\partial X\) of dimension \(\delta(\Gamma)\) and extends them in natural ways to obtain \(\Gamma\)-invariant measures \(\widetilde{\mu}\) on \(\mathfrak{H}\) and \(m_{\Gamma}\) on \(SX\). The measure \(m_{\Gamma}\) on \(SX\) is the Bowen-Margulis-Sullivan (BMS) measure and is supported on \(\Lambda(\Gamma)\).

The main focus of the article is on groups \(\Gamma\) of divergence type whose closed geodesic length spectrum is nonarithmetic (i.e., generates a dense subgroup of \(\mathbb{R}\)). Under these conditions the action of \(\Gamma\) on \(\{\mathfrak{H}, \widetilde{\mu} \}\) is ergodic. Some of the strongest results hold when the BMS measure \(m_{\Gamma}\) on \(SX\) for a given conformal density \(\mu\) is finite. For example, the geodesic flow is mixing of order \(p\) on \(\Lambda(\Gamma)\times \Lambda(\Gamma)\times \mathbb{R}\) for any positive integer \(p\). Moreover the measures \(\{\mu_{x} : x \in X \}\) are weak limits of weighted sums of Dirac measures concentrated on the orbits of \(\Gamma\) in \(X\). The normalized BMS measure in \(SX\) is also a weak limit of weighted sums of measures concentrated on closed orbits of the geodesic flow in \(SX\). One can also describe the space average of an \(L^{1}\) function \(\varphi\) on \(SX/\Gamma\) as a suitable limit as r \(\rightarrow \infty\) of averages of \(\varphi\) over balls of radius \(r\) of almost any horosphere of \(X\). Finally, one can characterize the natural measure \(\widetilde{\mu}\) on \(\mathfrak{H}\) up to scaling as the unique \(\Gamma\)-invariant measure supported on \(\Lambda_{c}(\Gamma)\times \mathbb{R} \subset \partial X\times\mathbb{R}\), where \(\Lambda_{c}(\Gamma) \subset \Lambda(\Gamma)\) is the conical limit set of \(\Gamma\). This result actually requires an additional condition on \(\Lambda_{c}(\Gamma)\) that is satisfied by almost all known examples.

The main focus of the article is on groups \(\Gamma\) of divergence type whose closed geodesic length spectrum is nonarithmetic (i.e., generates a dense subgroup of \(\mathbb{R}\)). Under these conditions the action of \(\Gamma\) on \(\{\mathfrak{H}, \widetilde{\mu} \}\) is ergodic. Some of the strongest results hold when the BMS measure \(m_{\Gamma}\) on \(SX\) for a given conformal density \(\mu\) is finite. For example, the geodesic flow is mixing of order \(p\) on \(\Lambda(\Gamma)\times \Lambda(\Gamma)\times \mathbb{R}\) for any positive integer \(p\). Moreover the measures \(\{\mu_{x} : x \in X \}\) are weak limits of weighted sums of Dirac measures concentrated on the orbits of \(\Gamma\) in \(X\). The normalized BMS measure in \(SX\) is also a weak limit of weighted sums of measures concentrated on closed orbits of the geodesic flow in \(SX\). One can also describe the space average of an \(L^{1}\) function \(\varphi\) on \(SX/\Gamma\) as a suitable limit as r \(\rightarrow \infty\) of averages of \(\varphi\) over balls of radius \(r\) of almost any horosphere of \(X\). Finally, one can characterize the natural measure \(\widetilde{\mu}\) on \(\mathfrak{H}\) up to scaling as the unique \(\Gamma\)-invariant measure supported on \(\Lambda_{c}(\Gamma)\times \mathbb{R} \subset \partial X\times\mathbb{R}\), where \(\Lambda_{c}(\Gamma) \subset \Lambda(\Gamma)\) is the conical limit set of \(\Gamma\). This result actually requires an additional condition on \(\Lambda_{c}(\Gamma)\) that is satisfied by almost all known examples.

Reviewer: Patrick Eberlein (Chapel Hill)

### MSC:

37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |

37F35 | Conformal densities and Hausdorff dimension for holomorphic dynamical systems |

37A25 | Ergodicity, mixing, rates of mixing |

37C40 | Smooth ergodic theory, invariant measures for smooth dynamical systems |

28D10 | One-parameter continuous families of measure-preserving transformations |

53C22 | Geodesics in global differential geometry |

53D25 | Geodesic flows in symplectic geometry and contact geometry |

57R30 | Foliations in differential topology; geometric theory |