Pure Maths Colloquium: Dimensions of sets arising from iterated function systems -- with a special emphasis on self-affine sets
7 August 2017 16:00 in CM221
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 uniformly shrunk copies of itself) to the theory of self-affine sets (where shrinking is not uniform).
One of the most important results in the dimension theory of self-affine sets is a result of Falconer from 1988. His 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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