The $L^q$-spectrum for a class of self-similar measures with overlap

Hare, Kathryn E.; Shen, Wanchun

doi:10.4310/ajm.2021.v25.n2.a3

Cited by 3 publications

(5 citation statements)

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“…While the L q -spectrum contains a point of non-differentiability at some q 0 < 0, the measure still satisfies the multifractal formalism [9]. In this case, and other related special cases, recent work of Hare, Hare, and Shen [20] explains this phenomenon in a combinatorial way. However, in general, more work is needed to understand this phenomenon.…”

Section: Introductionmentioning

confidence: 84%

“…In this case, it was observed that the set of attainable local dimensions is not an interval and the multifractal formalism fails [23]. The problem here is, in some sense, that the measure µ p is too small at certain points in K. This measure, and other related measures, were studied in detail [13,20,30,36] and a modified multifractal formalism was proven therein. In these cases, the failure occurs at some point q < 0.…”

Section: Introductionmentioning

confidence: 99%

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Geometric and Combinatorial Properties of Self-similar Multifractal Measures

Rutar

2020

Preprint

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For any self-similar measure µ in R, we show that the distribution of µ is controlled by products of non-negative matrices governed by a finite or countable graph depending only on the IFS. This generalizes the net interval construction of Feng from the equicontractive finite type case. When the measure satisfies the weak separation condition, we prove that this directed graph has a unique attractor. This allows us to verify the multifractal formalism for restrictions of µ to certain compact subsets of R, determined by the directed graph. When the measure satisfies the generalized finite type condition with respect to an open interval, the directed graph is finite and we prove that if the multifractal formalism fails for any q ∈ R, there must be a cycle with no vertices in the attractor. As a direct application, we verify the complete multifractal formalism for an uncountable family of IFSs with overlaps and without logarithmically commensurable contraction ratios.

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Section: Introductionmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

Geometric and Combinatorial Properties of Self-similar Multifractal Measures

Rutar

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…In this case, it was observed that the set of attainable local dimensions is not an interval and the multifractal formalism fails [24]. The problem here is, in some sense, that the measure μ p is too small at certain points in K. This measure, and other related measures, were studied in detail [14,21,31,38] and a modified multifractal formalism was proven therein. In these cases, the failure occurs at some point q < 0.…”

Section: A Rutarmentioning

confidence: 98%

“…The problem here is, in some sense, that the measure is too small at certain points in K . This measure, and other related measures, were studied in detail [14, 21, 31, 38] and a modified multifractal formalism was proven therein. In these cases, the failure occurs at some point .…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Geometric and combinatorial properties of self-similar multifractal measures

Rutar

2022

Ergod. Th. Dynam. Sys.

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For any self-similar measure $\mu $ in $\mathbb {R}$ , we show that the distribution of $\mu $ is controlled by products of non-negative matrices governed by a finite or countable graph depending only on the iterated function system of similarities (IFS). This generalizes the net interval construction of Feng from the equicontractive finite-type case. When the measure satisfies the weak separation condition, we prove that this directed graph has a unique attractor. This allows us to verify the multifractal formalism for restrictions of $\mu $ to certain compact subsets of $\mathbb {R}$ , determined by the directed graph. When the measure satisfies the generalized finite-type condition with respect to an open interval, the directed graph is finite and we prove that if the multifractal formalism fails at some $q\in \mathbb {R}$ , there must be a cycle with no vertices in the attractor. As a direct application, we verify the complete multifractal formalism for an uncountable family of IFSs with exact overlaps and without logarithmically commensurable contraction ratios.

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Matrix representations for some self-similar measures on $${\mathbb {R}}^{d}$$

2022

Math. Z.

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The $L^q$-spectrum for a class of self-similar measures with overlap

Cited by 3 publications

References 0 publications

Geometric and Combinatorial Properties of Self-similar Multifractal Measures

Geometric and Combinatorial Properties of Self-similar Multifractal Measures

Geometric and combinatorial properties of self-similar multifractal measures

Matrix representations for some self-similar measures on $${\mathbb {R}}^{d}$$

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