Dimension of self-affine sets for fixed translation vectors
Bárány, B., Käenmäki, A., & Koivusalo, H. (2018). Dimension of self-affine sets for fixed translation vectors. Journal of the London Mathematical Society, 98 (1), 223-252. doi:10.1112/jlms.12132
Published inJournal of the London Mathematical Society
© 2018 London Mathematical Society
An affine iterated function system is a finite collection of affine invertible contractions and the invariant set associated to the mappings is called self-affine. In 1988, Falconer proved that, for given matrices, the Hausdorff dimension of the self-affine set is the affinity dimension for Lebesgue almost every translation vectors. Similar statement was proven by Jordan, Pollicott, and Simon in 2007 for the dimension of self-affine measures. In this article, we have an orthogonal approach. We introduce a class of self-affine systems in which, given translation vectors, we get the same results for Lebesgue almost all matrices. The proofs rely on Ledrappier-Young theory that was recently verified for affine iterated function systems by Bárány and Käenmäki, and a new transversality condition, and in particular they do not depend on properties of the Furstenberg measure. This allows our results to hold for self-affine sets and measures in any Euclidean space.