Modulus and Poincaré Inequalities on Non-Self-Similar Sierpiński Carpets

Mackay, John; Tyson, Jeremy T.; Wildrick, Kevin

doi:10.1007/s00039-013-0227-6

Cited by 29 publications

(54 citation statements)

References 32 publications

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“…We recall a construction studied in [, Section 1]. Let

a = (a_{1}, a_{2}, \dots)

be a sequence of positive reals such that for any i the number

\frac{1}{a_{i}}

is an odd integer strictly greater than one.…”

Section: Sierpinski Carpets Of Positive Two‐dimensional Lebesgue Measurementioning

confidence: 99%

“…Then any nonempty open subset of

S_{a}

is topologically one‐dimensional. If

a \in l^{2}

, then any nonempty open subset of

S_{a}

has positive two‐dimensional Lebesgue measure λ 2 , see [, Proposition 3.1 (iv)], and the restriction of λ 2 to

S_{a}

is Ahlfors 2‐regular. Examples In Figure on the right hand side we have

a_{n} = \frac{1}{2 n + 1} \in l^{2}

, and so any nonempty open subset of the carpet

S_{a}

associated with this sequence

a = {(a_{n})}_{n \geq 1}

has positive λ 2 measure.…”

Section: Sierpinski Carpets Of Positive Two‐dimensional Lebesgue Measurementioning

confidence: 99%

“…On the left, the standard self‐similar Sierpinski carpet. On the right, a non‐self‐similar Sierpinski carpet

S_{false(1 / 3, 1 / 5, 1 / 7, \dots false)}

from the paper of Mackay, Tyson and Wildrick, (see p. 3 of arXiv:1201.3548 for the precise definition)…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, if the holes in the carpet are not too small, then [36, Theorem 4.2] applies (see Remark 4.9 below) and tells that within the 2 -space of 1-forms F I G U R E 1 On the left, the standard self-similar Sierpinski carpet. On the right, a non-self-similar Sierpinski carpet (1∕3,1∕5,1∕7,… ) from the paper of Mackay, Tyson and Wildrick, [48] (see p. 3 of arXiv:1201.3548 for the precise definition) the locally harmonic 1-forms are dense in the orthogonal complement of the exact 1-forms. Here we call a square integrable 1-form locally harmonic if we can find a finite open cover { } ∈ of and functions ℎ , ∈ , harmonic in the Dirichlet form sense, such that for any ∈ we have = ℎ , where denotes the exterior derivation.…”

Section: Introductionmentioning

confidence: 99%

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Densely defined non‐closable curl on carpet‐like metric measure spaces

Hinz

Teplyaev

2018

Mathematische Nachrichten

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The paper deals with the possibly degenerate behaviour of the exterior derivative operator defined on 1‐forms on metric measure spaces. The main examples we consider are the non self‐similar Sierpinski carpets recently introduced by Mackay, Tyson and Wildrick. Although topologically one‐dimensional, they may have positive two‐dimensional Lebesgue measure and carry nontrivial 2‐forms. We prove that in this case the curl operator (and therefore also the exterior derivative on 1‐forms) is not closable, and that its adjoint operator has a trivial domain. We also formulate a similar more abstract result. It states that for spaces that are, in a certain way, structurally similar to Sierpinski carpets, the exterior derivative operator taking 1‐forms into 2‐forms cannot be closable if the martingale dimension is larger than one.

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“…We recall a construction studied in [, Section 1]. Let

a = (a_{1}, a_{2}, \dots)

be a sequence of positive reals such that for any i the number

\frac{1}{a_{i}}

is an odd integer strictly greater than one.…”

Section: Sierpinski Carpets Of Positive Two‐dimensional Lebesgue Measurementioning

confidence: 99%

“…Then any nonempty open subset of

S_{a}

is topologically one‐dimensional. If

a \in l^{2}

, then any nonempty open subset of

S_{a}

has positive two‐dimensional Lebesgue measure λ 2 , see [, Proposition 3.1 (iv)], and the restriction of λ 2 to

S_{a}

is Ahlfors 2‐regular. Examples In Figure on the right hand side we have

a_{n} = \frac{1}{2 n + 1} \in l^{2}

, and so any nonempty open subset of the carpet

S_{a}

associated with this sequence

a = {(a_{n})}_{n \geq 1}

has positive λ 2 measure.…”

Section: Sierpinski Carpets Of Positive Two‐dimensional Lebesgue Measurementioning

confidence: 99%

“…On the left, the standard self‐similar Sierpinski carpet. On the right, a non‐self‐similar Sierpinski carpet

S_{false(1 / 3, 1 / 5, 1 / 7, \dots false)}

from the paper of Mackay, Tyson and Wildrick, (see p. 3 of arXiv:1201.3548 for the precise definition)…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Densely defined non‐closable curl on carpet‐like metric measure spaces

Hinz

Teplyaev

2018

Mathematische Nachrichten

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“…The construction in question undergoes by dividing each of the level n squares into a 2 n subsquares in an obvious manner and removing the middle square of the level n + 1 from each of the level n squares. In [MTW13] these carpets were studied in connection with Poincaré inequalities. Mackay et al showed that (S a , d, µ) supports a p-Poincaré inequality for p > 1 if and only if a ∈ ℓ 2 .…”

Section: Introductionmentioning

confidence: 99%

On (alpha)_n-regular sets

Ojala¹

2014

Ann. Acad. Sci. Fenn. Math.

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Abstract. We define (α n )-regular sets in uniformly perfect metric spaces. This definition is quasisymmetrically invariant and the construction resembles generalized dyadic cubes in metric spaces. For these sets we then determine the necessary and sufficient conditions to be fat (or thin). In addition we discuss restrictions of doubling measures to these sets, and, in particular, give a sufficient condition to retain at least some of the restricted measures doubling on the set. Our main result generalizes and extends analogous results that were previously known to hold on the real line.

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Thin Loewner Carpets and Their Quasisymmetric Embeddings in S²

Cheeger

Eriksson-Bique

2021

Comm Pure Appl Math

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A carpet is a metric space that is homeomorphic to the standard Sierpiński carpet in ℝ2, or equivalently, in S2. A carpet is called thin if its Hausdorff dimension is <2. A metric space is called Q‐Loewner if its Q‐dimensional Hausdorff measure is Q‐Ahlfors regular and if it satisfies a ()1,Q‐Poincaré inequality. As we will show, Q‐Loewner planar metric spaces are always carpets, and admit quasisymmetric embeddings into the plane. In this paper, for every pair ()Q,Q′, with 1

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Modulus and Poincaré Inequalities on Non-Self-Similar Sierpiński Carpets

Cited by 29 publications

References 32 publications

Densely defined non‐closable curl on carpet‐like metric measure spaces

Densely defined non‐closable curl on carpet‐like metric measure spaces

On (alpha)_n-regular sets

Thin Loewner Carpets and Their Quasisymmetric Embeddings in S²

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Modulus and Poincaré Inequalities on Non-Self-Similar Sierpiński Carpets

Cited by 29 publications

References 32 publications

Densely defined non‐closable curl on carpet‐like metric measure spaces

Densely defined non‐closable curl on carpet‐like metric measure spaces

On (alpha)_n-regular sets

Thin Loewner Carpets and Their Quasisymmetric Embeddings in S2

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Product

Resources

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Thin Loewner Carpets and Their Quasisymmetric Embeddings in S²