On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems

TL;DR

This paper establishes conditions under which the Hausdorff and affinity dimensions of planar self-affine sets are equal, using self-affine measures on positive subsystems and recent advances in the field.

Contribution

It demonstrates that the affinity dimension equals the supremum of Lyapunov dimensions on certain positive subsystems, leading to new criteria for dimension equality without domination assumptions.

Findings

Equality of Hausdorff and affinity dimensions under mild conditions

Construction of explicit examples with dimension equality without domination

New criteria linking self-affine measures and set dimensions

Abstract

Under mild conditions we show that the affinity dimension of a planar self-affine set is equal to the supremum of the Lyapunov dimensions of self-affine measures supported on self-affine proper subsets of the original set. These self-affine subsets may be chosen so as to have stronger separation properties and in such a way that the linear parts of their affinities are positive matrices. Combining this result with some recent breakthroughs in the study of self-affine measures and their associated Furstenberg measures, we obtain new criteria under which the Hausdorff dimension of a self-affine set equals its affinity dimension. For example, applying recent results of B\'{a}r\'{a}ny, Hochman-Solomyak and Rapaport, we provide new explicit examples of self-affine sets whose Hausdorff dimension equals its affinity dimension, and for which the linear parts do not satisfy any domination assumptions.