Local dimensions of self-similar measures satisfying the finite neighbour condition

Kathryn Hare; Alex Rutar

doi:10.1088/1361-6544/ac8040

Local dimensions of self-similar measures satisfying the finite neighbour condition

Kathryn Hare, Alex Rutar^*

^*Corresponding author for this work

Pure Mathematics

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	4876-4904
Number of pages	29
Journal	Nonlinearity
Volume	35
Issue number	9
DOIs	https://doi.org/10.1088/1361-6544/ac8040
Publication status	Published - 11 Aug 2022

Keywords

Iterated function system
Self-similar
Local dimension
Multifractal analysis
Weak separatopm condition

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title = "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.",

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author = "Kathryn Hare and Alex Rutar",

note = "KEH was supported by NSERC Grant 2016-03719. AR was supported by this grant as well as EPSRC Grant EP/V520123/1.",

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T1 - Local dimensions of self-similar measures satisfying the finite neighbour condition

AU - Hare, Kathryn

AU - Rutar, Alex

N1 - KEH was supported by NSERC Grant 2016-03719. AR was supported by this grant as well as EPSRC Grant EP/V520123/1.

PY - 2022/8/11

Y1 - 2022/8/11

N2 - 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.

AB - 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.

KW - Iterated function system

KW - Self-similar

KW - Local dimension

KW - Multifractal analysis

KW - Weak separatopm condition

U2 - 10.1088/1361-6544/ac8040

DO - 10.1088/1361-6544/ac8040

M3 - Article

SN - 0951-7715

VL - 35

SP - 4876

EP - 4904

JO - Nonlinearity

JF - Nonlinearity

IS - 9

ER -

Local dimensions of self-similar measures satisfying the finite neighbour condition

Abstract

Keywords

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