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Transactions of the American Mathematical Society

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

Authors: Ian D. Morris and Pablo Shmerkin
Journal: Trans. Amer. Math. Soc. 371 (2019), 1547-1582
MSC (2010): Primary 28A80, 37C45; Secondary 37D35
Published electronically: October 17, 2018
MathSciNet review: 3894027
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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 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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Additional Information

Ian D. Morris
Affiliation: Department of Mathematics, University of Surrey, Guildford GU2 7XH, United Kingdom

Pablo Shmerkin
Affiliation: Departamento de Matemáticas y Estadísticas and CONICET, Universidad Torcuato Di Tella, Av. Figueroa Alcorta 7350 (C1428BCW), Buenos Aires, Argentina

Received by editor(s): December 29, 2016
Received by editor(s) in revised form: June 16, 2017
Published electronically: October 17, 2018
Additional Notes: The first author was supported by the Engineering and Physical Sciences Research Council (grant number EP/L026953/1).
The second author was partially supported by project PICT 2013-1393 (ANPCyT)
Article copyright: © Copyright 2018 American Mathematical Society