Local dimensions of self-similar measures satisfying the finite neighbour condition
Abstract
We study sets of local dimensions for self-similar measures in R satisfying the finite neighbour condition, which is formally stronger than the weak separation condition (WSC) but satisfied in all known examples. Under a mild technical assumption, we establish that the set of attainable local dimensions is a finite union of (possibly singleton) compact intervals. The number of intervals is bounded above by the number of non-trivial maximal strongly connected components of a finite directed graph construction depending only on the governing iterated function system. We also explain how our results allow computations of the sets of local dimensions in many explicit cases. This contextualises and generalises a vast amount of prior work on sets of local dimensions for self-similar measures satisfying the WSC.
Citation
Hare , K & Rutar , A 2022 , ' Local dimensions of self-similar measures satisfying the finite neighbour condition ' , Nonlinearity , vol. 35 , no. 9 , pp. 4876-4904 . https://doi.org/10.1088/1361-6544/ac8040
Publication
Nonlinearity
Status
Peer reviewed
ISSN
0951-7715Type
Journal article
Rights
Copyright © 2022 IOP Publishing Ltd & London Mathematical Society. Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Description
KEH was supported by NSERC Grant 2016-03719. AR was supported by this grant as well as EPSRC Grant EP/V520123/1.Collections
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