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Documents  28A80 | enregistrements trouvés : 3

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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. 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 ...

28A80 ; 37A10 ; 03D99 ; 54H20

Multi angle  Curvature measures of random sets
Zähle, Martina (Auteur de la Conférence) | CIRM (Editeur )

A survey on some developments in curvature theory for random sets will be given. We first consider previous models with classical singularities like polyconvex sets or unions of sets with positive reach. The main part of the talk concerns extensions to certain classes of random fractals which have been investigated in the last years. In these cases limits of rescaled versions for suitable approximations are used.

53C65 ; 52A22 ; 60D05 ; 28A80 ; 28A75

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