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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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##### References:
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