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Divisible convex sets. (Convexes divisibles.) (French. English summary) Zbl 1010.37014
Summary: We study the discrete groups \(\Gamma\) of projective transformations of the real projective space which divide a properly convex open subset \(\Omega\) (i.e., which preserve \(\Omega\) and whose quotient \(\Gamma \setminus \Omega\) is compact). We describe the Zariski closure of these groups \(\Gamma\) and we study their “deformation space”. Suppose now that \(\Omega\) is strictly convex. We show that \(\Gamma\) is hyperbolic, that the geodesic flow of the quotient space \(\Gamma\setminus \Omega\) is Anosov, that the boundary \(\partial\Omega\) is of class \(C^1\) and that its normal map is Hölder continuous; moreover the following rigidity assertion holds: the normal map is absolutely continuous if and only if \(\Omega\) is an ellipsoid.

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
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