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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.

##### MSC:
 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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