Local dimensions of self-similar measures satisfying the finite neighbour condition*KEH was supported by NSERC Grant 2016-03719. AR was supported by this grant as well as EPSRC Grant EP/V520123/1. (1st September 2022)

• Record Type: Journal Article
• Authors: Hare, Kathryn E; Rutar, Alex
• Is Part Of: Nonlinearity. Volume 35:Number 9(2022)
• Page Start: 4876
• Page End: 4904
• Publication Date: 2022-09-01
• Subjects: iterated function system -- self-similar -- local dimension -- multifractal analysis -- weak separation condition; 28A80; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ac8040
• Languages: English
• ISSNs: 0951-7715

Abstract

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.