Dimension of self-affine sets for fixed translation vectors

Fractal Geometry and Dynamics

12 September 14:00 - 14:50

Henna Koivusalo - University of Vienna

In 1988, Falconer proved that, for a fixed collection of matrices, the Hausdorff dimension of the corresponding self-affine set is the affinity dimension for Lebesgue almost every choice of translation vectors. Similar statement was proved by Jordan, Pollicott, and Simon in 2007 for the dimension of self-affine measures. I discuss an orthogonal approach, introducing a class of affine iterated function systems in which, given translation vectors, for Lebesgue almost all matrices, the dimension of the corresponding self-affine set is the affinity dimension. Our proofs rely on Ledrappier-Young theory that was recently verified for affine iterated function systems by Barany and Kaenmaki, and a new transversality condition. In particular our argument does not directly depend on properties of the Furstenberg measure which allows our results to hold for self-affine sets and measures in any Euclidean space and not just in the plane. The work is joint with Balazs Barany and Antti Kaenmaki.
Kenneth J. Falconer
University of St Andrews
Maarit Järvenpää
University of Oulu
Antti Kupiainen
University of Helsinki
Francois Ledrappier
University of Notre Dame
Pertti Mattila
University of Helsinki


Maarit Järvenpää


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