University of Surrey

Test tubes in the lab Research in the ATI Dance Research

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.

[img]
Preview
Text
self-affine revised version with example.pdf - Accepted version Manuscript

Download (1MB) | Preview

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 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 :
NameEmailORCID
Morris, Iani.morris@surrey.ac.ukUNSPECIFIED
Shmerkin, PUNSPECIFIEDUNSPECIFIED
Date : 28 October 2017
Copyright Disclaimer : First published in Transactions of the American Mathematical Society in 2017, published by the American Mathematical Society.
Depositing User : Melanie Hughes
Date Deposited : 28 Jul 2017 13:19
Last Modified : 28 Jul 2017 13:19
URI: http://epubs.surrey.ac.uk/id/eprint/841781

Actions (login required)

View Item View Item

Downloads

Downloads per month over past year


Information about this web site

© The University of Surrey, Guildford, Surrey, GU2 7XH, United Kingdom.
+44 (0)1483 300800