Abstract
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.
Original language | English |
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Pages (from-to) | 223-252 |
Number of pages | 30 |
Journal | Journal of the London Mathematical Society |
Volume | 98 |
Issue number | 1 |
DOIs | |
Publication status | Published - 23 Apr 2018 |
Keywords
- 37C45 (primary)
- 28A80 (secondary)
- Self-affine set
- self-affine measure
- Hausdorff dimension