On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems
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.
Keywords
Cite
@article{arxiv.1602.08789,
title = {On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems},
author = {Ian D. Morris and Pablo Shmerkin},
journal= {arXiv preprint arXiv:1602.08789},
year = {2019}
}
Comments
v3: added several examples and figures, corrected statement of Proposition 7.2. v2: Added some new examples to show that the SOSC cannot be relaxed to the OSC in Theorems 1.2 and 1.3. Applications of some results of Hochman-Solomyak have been updated to more accurately reflect the form which those results take in Hochman and Solomyak's recent preprint. Minor additional typographical fixes