Dimensions of sets arising from iterated function systems -- with a special emphasis on self-affine sets

Henna Koivusalo (University of Vienna)

Tuesday 15th August, 2017 16:00-17:00 Maths Seminar Room (level 3)


In this colloquium style talk I will review the history of calculating 
dimensions of sets that arise as invariant sets of iterated function 
systems. I will, in particular, compare the theory of self-similar sets 
(where the set is a union of shrunk copies of itself) to the theory of 
self-affine sets (where the set is a union of affine images of itself).

One of the most important results in the dimension theory of self-affine 
sets is a result of Falconer from 1988. He showed that Lebesgue almost 
surely, the dimension of a self-affine set does not depend on 
translations of the pieces of the set. A similar statement was proven by 
Jordan, Pollicott, and Simon in 2007 for the dimension of self-affine 
measures. At the end of my talk I will explain an orthogonal approach to 
the dimension calculation, introducing a class of self-affine systems in 
which, given translations, a dimension result holds for Lebesgue almost 
all choices of deformations.

This work is joint with Balazs Barany and Antti Kaenmaki.

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