Dimension of self-affine sets for fixed translation vectors

TL;DR

This paper introduces a new approach to understanding the dimension of self-affine sets and measures, using Ledrappier-Young theory and a transversality condition, applicable in any Euclidean space.

Contribution

It presents an orthogonal method that establishes dimension results for self-affine sets and measures for almost all matrices, independent of Furstenberg measure properties.

Findings

Results hold for Lebesgue almost all matrices given translation vectors.

The approach applies to self-affine sets and measures in any Euclidean space.

Uses Ledrappier-Young theory and a new transversality condition.

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.