6533b82afe1ef96bd128cc9f

RESEARCH PRODUCT

Dimension of self-affine sets for fixed translation vectors

Balazs BaranyAntti KaenmakiHenna Koivusalo

subject

Self-affine setvektoritself-affine measurevectorsmatematiikka37C45 28A80FOS: MathematicsHausdorff dimensionDynamical Systems (math.DS)Mathematics - Dynamical Systems37C45 (primary)28A80 (secondary)matemaattiset objektit

description

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 B\'ar\'any and K\"aenm\"aki, 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.

yearjournalcountryeditionlanguage
2016-11-28
https://dx.doi.org/10.48550/arxiv.1611.09196