Dimension of self-affine sets for fixed translation vectors

Balazs Barany*, Antti Kaenmaki*, Henna L L Koivusalo*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

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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 languageEnglish
Pages (from-to)223-252
Number of pages30
JournalJournal of the London Mathematical Society
Volume98
Issue number1
DOIs
Publication statusPublished - 23 Apr 2018

Keywords

  • 37C45 (primary)
  • 28A80 (secondary)
  • Self-affine set
  • self-affine measure
  • Hausdorff dimension

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