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An Anosov action on the bundle of Weyl chambers. (English) Zbl 0555.58023

The geodesic flow of a Riemannian symmetric space of non-compact type is an Anosov flow if and only if the rank of the space is one. In this note an action is introduced which is Anosov for all Riemannian symmetric spaces of non-compact type. The action consists of parallel translating Weyl chambers within their supporting totally geodesic flat subspaces. If the rank of the space is one, then the action coincides with the geodesic flow. For spaces of rank higher than one, it is no longer a flow, but an action of a higher-dimensional abelian Lie group. After a group- theoretical definition of the action and a proof of its Anosov character, geometrical interpretations are given in terms of horocycles.

MSC:

37D99 Dynamical systems with hyperbolic behavior
37C10 Dynamics induced by flows and semiflows
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