Sauf mention contraire le séminaire a lieu le lundi à 14h00 dans la salle 16 du bâtiment 22

Les exposés à venir

2 octobre 2017

Weak capacity in Ahlfors regular metric spaces.

Jeff Lindquist (University of Helsinki)

We construct and use a hyperbolic filling (similar to the Cayley graph of a finitely generated group) of a $Q$-regular compact doubling metric space $Z$ to define the notion of the weak $p$-capacity between appropriate subsets of $Z$. This notion extends modulus and is preserved up to constants by quasisymmetric maps. We explore some applications involving conformal dimension and quasisymmetric uniformization of metric $2$-spheres.

9 octobre 2017

TBA

Anthony Genevois (Aix-Marseille Université)

6 novembre 2017

Royal Measures and the Feldman Katok Pseudometric

Martha Łącka (Jagiellonian University, Kraków)

The GIKN construction was introduced by Gorodetski, Ilyashenko, Kleptsyn, and Nalsky in [Functional Analysis and its Applications, 39 (2005), 21--30]. It gives a nonhyperbolic ergodic measure which is a weak$^*$ limit of a special sequence of measures supported on periodic orbits. This method was later adapted by numerous authors (Bonatti, Cheng, Crovisier, Diaz, Gan, Wang, Yang, Zhang) and provided examples of nonhyperbolic invariant measures in various settings. We prove that the result of the GIKN construction is always a loosely Kronecker measure in the sense of Ornstein, Rudolph, and Weiss (equivalently, standard measure in the sense of Katok, another name is loosely Bernoulli measure with zero entropy). For a proof we introduce and study the Feldman-Katok pseudometric $\bar F$. The pseudodistance $\bar F$ is a topological counterpart of the f-bar metric for finite-state stationary stochastic processes introduced by Feldman and, independently, by Katok, later developed by Ornstein, Rudolph, and Weiss. We show that every measure given by the GIKN construction is the $\bar F$-limit of a sequence of periodic measures. On the other hand we prove that a measure which is the $\bar F$-limit of a sequence of ergodic measures is ergodic and its entropy is smaller or equal than the lower limit of entropies of measures in the sequence. Furthermore we demonstrate that $\bar F$-Cauchy sequence of periodic measures tends in the weak$^*$ topology to a loosely Kronecker measure. The talk will be based on a joint work with Dominik Kwietniak.

13 novembre 2017

Soutenance

TBA

Axel Rogue (Université de Rennes 1)

TBA

20 novembre 2017

TBA

John Mackay (University of Bristol)

TBA

27 novembre 2017

TBA

Bassam Fayad (Institut de Mathématiques de Jussieu)

TBA

11 decembre 2017

TBA

Katrin Gelfert (Universidade Federal do Rio de Janeiro)

TBA

Les exposés passés

25 septembre 2017

Critical exponents for normal subgroups

Dougall, Rhiannon (Université de Nantes)

Fix a cocompact group $\Gamma_0$ of isometries of a negatively curved, simply connected space $X$. We are interested in the dynamics of it's normal subgroups $\Gamma$. Namely, we study the critical exponent $\delta_\Gamma$, which is the exponential growth rate of the $\Gamma$-orbit of a point. We characterise the existence of a gap $\delta_\Gamma<\delta_{\Gamma_0}$ uniform in a family of normal subgroups $\Gamma$, in terms of permutation representations given by the quotients $\Gamma_0/\Gamma$. The proof uses the symbolic dynamics for the geodesic flow, for which we obtain the analogous statements for countable state shifts obtained as group extensions of a finite state shift.

18 septembre 2017

Upgrading fixed points without Bounded Generation

Mimura, Masato (EPFL)

It is well-known that $SL(n,\mathbb Z)$ for $n$ at least $3$ has Kazhdan's property (T). By the Delorme--Guichardet theorem, this property is equivalent to saying that every group action on a Hilbert space by affine isometries admits a global fixed point. In this talk, I will present the "most down-to-earth" proof of this fact: we do not appeal to either of the facts that $SL(n,\mathbb Z)$ is a lattice in $SL(n,\mathbb R)$, or that it is boundedly generated by certain subgroups. This proof enables us to prove fixed point properties for a much wider class of groups, along exactly the same line.

11 septembre 2017

Sur les propriétés asymptotiques des groupes linéaires

Sert, Çağrı (ETHZ)

Cet exposé portera sur quelques aspects probabilistes et déterministes de l’étude asymptotique des groupes linéaires. Plus précisément, on se concentrera aux sous-groupes Zariski denses de $G=SL(d,\mathbb R)$ et après avoir rappelé les résultats fondamentaux (de Furstenberg, Kesten, Le Page, Guivarc’h, Raugi, Goldsheid, Margulis, Benoist, Quint etc.), on parlera des probabilités de grandes déviations et discutera l’homologue du théorème de Cramer sur le principe de grandes déviations. Pour préciser ce dernier résultat, dans un second temps, on parlera d’une notion d’ensemble limite déterministe intimement liée au cône limite de Benoist et la notion de rayon spectral joint dont on rappellera les définitions. Finalement, on parlera de zone de Lyapunov et énoncera des questions ouvertes. (Travaux en partie communs avec Emmanuel Breuillard)