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arXiv · 1602.08789

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

Ian D. Morris·Pablo Shmerkin
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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.

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Ian D. Morris, Pablo Shmerkin. 2016-02-29. On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems. https://arxiv.org/abs/1602.08789

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