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

Morris, Ian and Shmerkin, P (2017) On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems Transactions of the American Mathematical Society.

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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 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.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
Shmerkin, P
Date : 28 October 2017
DOI : 10.1090/tran/7119
Copyright Disclaimer : First published in Transactions of the American Mathematical Society in 2017, published by the American Mathematical Society.
Related URLs :
Depositing User : Melanie Hughes
Date Deposited : 28 Jul 2017 13:19
Last Modified : 16 Jan 2019 18:54

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