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Topological mixing in \(\text{Cat} (-1)\)-spaces. (English) Zbl 0979.57002
The authors consider a metric space \(X\) which is proper and which satisfies the \(\text{CAT}(-1)\) comparison property, and a non-elementary discrete group \(\Gamma\) of isometries of \(X\) which acts properly discontinuously, and they study dynamical properties of the geodesic flow on the space of geodesics \(GY\) associated to the quotient space \(Y=X/\Gamma\). The geodesic flow on \(GY\) is said to be topologically mixing if given any open sets \({\mathcal O}\) and \({\mathcal U}\) in \(GY\), there exists a real number \(t_0>0\) such that for all \(t\) with \(|t|\geq t_0\), we have \(t. {\mathcal O}\cup{\mathcal U}\not=\emptyset\). The authors show that the geodesic flow on \(GY\) is topologically mixing provided the two following properties hold : (i) \(\forall x,x'\in X\), there exists \(\xi\in\partial X\) such that \(\alpha(\xi,x,x')=0\), where \(\alpha\) is a generalized Busemann function on \((\partial X\cup X)\times X\times X\) which the authors define, (ii) the non-wandering set of the flow is equal to \(GY\). The authors note that special cases to which their result applies include the following : (A) compact negatively curved polyhedra, (B) compact quotients of proper geodesically complete \(\text{CAT}(-1)\)-spaces by a one-ended group of isometries, and (C) finite \(n\)-dimensional ideal polyhedra (that is, spaces obtained by gluing together along their boundaries ideal \(n\)-simplices of hyperbolic \(n\)-space).

MSC:
57M20 Two-dimensional complexes (manifolds) (MSC2010)
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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