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.