In the first part of this paper, we consider several natural problems about locally homogeneous rigid geometric structures. In particular, we formulate a notion of topological completeness which is adapted to the study of global rigidity of chaotic dynamical systems. In the second part of the paper, we prove the following result: let ? be a C ~∞ expanding map of a closed manifold. If ? preserves a topologically complete C ~∞ rigid geometric structure, then ? is C ~∞ conjugate to an expanding infra-nilendomorphism.
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