Yves Benoist

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Yves Benoist is a French mathematician who, among other things, deals with dynamics of groups in homogeneous spaces. He is Director of Research of the CNRS at the University of Paris-South .


In 1990, together with Patrick Foulon (* 1954) and François Labourie , he solved a long open conjecture about Anosov rivers on compact, negatively curved manifolds. In the 2000s he wrote a series of papers on the discrete automorphism group of certain open convex sets in projective space (and periodic tilings with these), e.g. B. conic sections.

In 2011 he received the Clay Research Award with his former PhD student Jean-François Quint . Her spectacular work on stationary measures and closed orbites for the effects of non-Abelian groups on homogeneous spaces was highlighted in the laudation, and especially her proof of a conjecture by Hillel Fürstenberg . They showed that in homogeneous spaces of finite volume, the orbits of a random walk with a Zariski - dense subset are equally distributed. As a simple example, they looked at the cats figure of Vladimir Arnold on the torus that maps it into itself. While rational points have finite orbits, irrational points have infinite but not necessarily evenly distributed ones. For the combination of a first cat image T with a suitably selected second cat image U, it follows from your theorem that, given a random sequence of the images T and U, the orbits of all irrational points are evenly distributed.

In 2012 he gave the Takagi Lecture in Kyoto at the Research Institute for Mathematical Sciences (RIMS). In 2014 he was invited speaker at the ICM in Seoul (Recurrence on the space of lattices).

Fanny Kassel is one of his doctoral students .


  • With N. de Saxcé: A spectral gap theorem in simple Lie groups. Invent. Math. 205 (2016), no.2, 337-361.
  • With J.-F. Quint: Mesures stationnaires et fermés invariants des espaces homogènes. I: Ann. of Math. (2) 174 (2011), no. 2, 1111-1162. II: J. Amer. Math. Soc. 26 (2013), no. 3, 659-734. III: Ann. of Math. (2) 178 (2013), no. 3, 1017-1059.
  • With J.-F. Quint: Random walks on finite volume homogeneous spaces. Invent. Math. 187 (2012), no. 1, 37-59.
  • Convexes divisibles.
    • I: Algebraic groups and arithmetic, 339-374, Tata Inst. Fund. Res., Mumbai, 2004.
    • II: Duke Math. J. 120 (2003), no. 1, 97-120.
    • III: Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 5, 793-832.
    • IV: Invent. Math. 164 (2006), no. 2, 249-278.
  • Propriétés asymptotiques des groupes linéaires. Geom. Funct. Anal. 7 (1997), no. 1, 1-47.
  • Actions propres sur les espaces homogènes reductifs. Ann. of Math. (2) 144 (1996) no. 2, 315-347.
  • With P. Foulon, F. Labourie: Flots d'Anosov à distributions stable et instable différentiables. J. Amer. Math. Soc. 5, no. 1, 33-74 (1992).


  • Jean-François Quint: Convexes divisibles, d'aprés Yves Benoist. Séminaire Bourbaki, June 2008.
  • François Ledrappier: Mesures stationnaires sur les espaces homogènes, d'après Yves Benoist et Jean-François Quint. Séminaire Bourbaki, June 2012.

Web links

Individual evidence

  1. J.-F. Quint: Convexes divisibles, d'apres Yves Benoist. Seminaire Bourbaki, No. 999, 2008.
  2. ^ Clay Research Award. 2011. Yves Benoist and Jean-François Quint. ( Memento of November 3, 2013 in the Internet Archive ).
  3. with and between 0 and 1
  4. Dynamique aléatoire. Description of your work at the CNRS.