Dimension of self‐affine sets for fixed translation vectors

Bárány, Balázs and Käenmäki, Antti and Koivusalo, Henna (2018) Dimension of self‐affine sets for fixed translation vectors. Journal of London Mathematical Society, 98 (1). pp. 223-252.

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Abstract

An affine iterated function system (IFS) 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 IFSs 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.

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