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Dimension of self-similar measures via additive combinatorics
Post-edited
Authors : Hochman, Mike (Author of the conference)
CIRM (Publisher )
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
CIRM (Publisher )
Abstract :
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.
MSC Codes : 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
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 27/01/14 Conference Date : 03/12/2013 Subseries : Research talks arXiv category : Dynamical Systems ; Analysis of PDEs ; Combinatorics Mathematical Area(s) : Combinatorics ; Analysis and its Applications ; Dynamical Systems & ODE Format : QuickTime (.mov) Video Time : 00:42:34 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2013-12-03_Hochman.mp4 
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Information on the Event
Event Title : Jean-Morlet Chair : Hyperbolicity and dimension / Chaire Jean-Morlet : Hyperbolicité et dimensionEvent Organizers : Hasselblatt, Boris ; Pesin, Yakov ; Schmeling, Joerg ; Troubetzkoy, Serge ; Vaienti, Sandro Dates : 02/12/13 - 06/12/13 Event Year : 2013 Event URL : https://www.chairejeanmorlet.com/1071.html
Citation Data
DOI : 10.24350/CIRM.V.18447603Cite 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 |
Bibliography
- Hochman, Michael . Self similar sets, entropy and additive combinatorics. arXiv:1307.6399 [math.CA] - http://arxiv.org/abs/1307.6399v3
- N. Aubrun and M. Sablik. Simulation of effective subshifts by two-dimensional subshifts. Acta Applicandae Mathematicae 126(1) (2013), 35-63 - http://dx.doi.org/10.1007/s10440-013-9808-5
- R. Berger. The undecidability of the domino problem. Mem. Amer. Math. Soc. 72(66) (1966). - http://ams.math.uni-bielefeld.de/mathscinet/search/publdoc.html?pg1=MR&s1=0216954&loc=fromreflist
