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

By Ian D. Morris, Pablo Shmerkin. Transactions of the American Mathematical Society, 2016

55 citations recorded by the source corpus.

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 Barany, 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/74e45b76060a2797d9995c71181afa6fe302bc60

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Citations in the corpus

Dimension of invariant measures for affine iterated function systems (2019, unknown): “[2, 3, 4, 5, 6, 7, 8, 9, 17, 25, 37, 50, 57, 59]).”
Hausdorff dimension of planar self-affine sets and measures (2017, methodology): “2 by the work of Morris-Shmerkin [33], who proved that under the hypotheses of Theorem 1.”