Matthieu JOSEPH



  • Products of snowflaked Euclidean lines are not minimal for looking down
    with T.Rajala, Anal. Geom. Metr. Spaces, 5 (2017), no. 1, 78-97.
    This work was published after a master research project in Jyväskylä (Finland), under the supervision of T. Rajala.
    We study a notion of self-similarity for metric spaces called BPI (Big Pieces of Itself). In order to classify all the different BPI geometries, David and Semmes introduced the notion of "looking down". In this article, we prove that the product of finitely many euclidean lines, all equipped with the distance (x,y)→|x-y|p for some p<1, is not minimal for looking down.


  • Continuum of allosteric actions for non-amenable surface groups
    Let Σ be a closed surface other than the sphere, the torus, the projective plane or the Klein bottle. We construct a continuum of p.m.p. ergodic minimal profinite actions for the fundamental group of Σ, that are topologically free but not essentially free, a property that we call allostery. Moreover, the IRS's we obtain are pairwise distinct.
  • Belinskaya's theorem is optimal
    with A. Carderi, F. Le Maître and R. Tessera
    Belinskaya's theorem states that given an ergodic measure-preserving transformation, any other transformation with the same orbits and an L1 cocycle must be flip-conjugate to it. Our main result shows that this theorem is optimal: for all p<1, the integrability condition on the cocycle cannot be relaxed to being in Lp. This also allows us to answer a question of Kerr and Li: for ergodic measure-preserving transformations, Shannon orbit equivalence doesn't boil down to flip-conjugacy.
  • Isometric orbit equivalence for probability-measure preserving actions
    We introduce the notion of isometric orbit equivalence for probability-measure preserving (pmp for short) actions . This notion asks the Schreier graphings defined by the actions of the groups to be isomorphic. First, we prove that the pmp actions of a group whose Cayley graph has a discret automorphisms group are rigid up to isometric orbit equivalence. In a second time, we introduct a method which enables us to prove that several groups admit pmp actions which are isometric orbit equivalent but not conjugate. This includes for instance free groups of finite rank with a free generating set.