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

Morris, Ian; Shmerkin, Pablo

doi:10.1090/tran/7334

On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems
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by Ian D. Morris and Pablo Shmerkin PDF

Trans. Amer. Math. Soc. 371 (2019), 1547-1582 Request permission

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 Bárány, Hochman–Solomyak, and Rapaport, we provide many new explicit examples of self-affine sets whose Hausdorff dimension equals its affinity dimension, and for which the linear parts do not satisfy any positivity or domination assumptions.

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