H 1 Dimension of self-similar measures via additive combinatorics

Auteurs : Hochman, Mike (Auteur de la Conférence)
CIRM (Editeur )

Loading the player...
introduction self-similar sets trivial bounds on dimension when are the trivial bounds achieved when is there inequality - exact overlaps exact overlapping conjecture self-similar measures almost exact overlapping theorem algebraic parameters Furstenberg's projection problem parametric families Bernoulli convolutions additive combinatorics

Résumé : I will discuss recent progress on understanding the dimension of self-similar sets and measures. The main conjecture in this field is that the only way that the dimension of such a fractal can be "non-full" is if the semigroup of contractions which define it is not free. The result I will discuss is that "non-full" dimension implies "almost non-freeness", in the sense that there are distinct words in the semigroup which are extremely close together (super-exponentially in their lengths). Applications include resolution of some conjectures of Furstenberg on the dimension of sumsets and, together with work of Shmerkin, progress on the absolute continuity of Bernoulli convolutions. The main new ingredient is a statement in additive combinatorics concerning the structure of measures whose entropy does not grow very much under convolution. If time permits I will discuss the analogous results in higher dimensions.

Codes MSC :
03D99 - None of the above but in this section
28A80 - Fractals
54H20 - Topological dynamics, See also {28Dxx, 34C35, 58Fxx}
37A10 - One-parameter continuous families of measure-preserving transformations

    Informations sur la Vidéo

    Langue : Anglais
    Date de publication : 27/01/14
    Date de captation : 03/12/13
    Collection : Research talks ; Analysis and its Applications ; Combinatorics ; Dynamical Systems and Ordinary Differential Equations
    Format : QuickTime (.mov) Durée : 00:42:34
    Domaine : Combinatorics ; Analysis and its Applications ; Dynamical Systems & ODE
    Audience : Chercheurs ; Doctorants , Post - Doctorants
    Download : https://videos.cirm-math.fr/2013-12-03_Hochman.mp4

Informations sur la rencontre

Nom de la rencontre : Jean-Morlet Chair : Hyperbolicity and dimension / Chaire Jean-Morlet : Hyperbolicité et dimension
Organisateurs de la rencontre : Hasselblatt, Boris ; Pesin, Yakov ; Schmeling, Joerg ; Troubetzkoy, Serge ; Vaienti, Sandro
Dates : 02/12/13 - 06/12/13
Année de la rencontre : 2013
URL Congrès : https://hasselblatttroubetzkoy.weebly.co...

Citation Data

DOI : 10.24350/CIRM.V.18447603
Cite this video as: Hochman, Mike (2013). Dimension of self-similar measures via additive combinatorics. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18447603
URI : http://dx.doi.org/10.24350/CIRM.V.18447603


Titres de périodiques et e-books électroniques (Depuis le CIRM)

Ressources Electroniques

Books & Print journals

Recherche avancée