Square Sierpiński carpets and Lattès maps

Bonk, Mario; Merenkov, Sergei

doi:10.1007/s00209-019-02435-1

Cited by 6 publications

(4 citation statements)

References 11 publications

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“…Due to the connection among the standard Sierpiński carpet, Julia sets and the Gromov hyperbolic groups, many papers are devoted to the study of the quasisymmetric geometry (note that a quasisymmetry is a homeomorphism with stronger requirements) of Sierpiński carpets. In [5,6], Bonk and Merenkov investigate the quasisymmetric homeomorphisms of the standard Sierpiński carpet. They show that every quasisymmetric homeomorphism of a GSC F = F (N, D) onto itself with N = 2m + 1 for some m ∈ Z + and D = {0, 1, .…”

Section: Theorem 13 ([27]mentioning

confidence: 99%

Connectedness and local cut points of generalized Sierpinski carpets

Dai¹,

Luo²,

Ruan³

et al. 2021

Preprint

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We investigate a homeomorphism problem on a class of self-similar sets called generalized Sierpiński carpets (or shortly GSCs). It follows from two well-known results by Hata and Whyburn that a connected GSC is homeomorphic to the standard Sierpiński carpet if and only if it has no local cut points. On the one hand, we show that to determine whether a given GSC is connected, it suffices to iterate the initial pattern twice. We also extend this result to higher dimensional cases and provide an effective method on how to draw the associated Hata graph which allows automatical detection by computer. On the other hand, we obtain a characterization of connected GSCs with local cut points. Toward this end, we first deal with connected GSCs with cut points and then return to the local problem. The former is achieved by dividing the collection of GSCs into two parts, i.e., fragile cases and non-fragile cases, and discussing them separately. An easilychecked necessary condition is presented in addition as a byproduct of the above discussion. Finally, we also look into the size of the associated digit sets and possible numbers of cut points of connected GSCs.

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Section: Theorem 13 ([27]mentioning

confidence: 99%

Connectedness and local cut points of generalized Sierpinski carpets

Dai¹,

Luo²,

Ruan³

et al. 2021

Preprint

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“…Resently, the transboundary extremal length(or transboundary modulus) has been applied in the study of the uniformization of metric spaces, see [3,5,6,10,21,22,23]. More results related to the Koebe uniformization problem can be found within the works that [4,11,13,14,20,28].…”

Section: Definition 11 ([25])mentioning

confidence: 99%

Front Matter

Wang

Fan

2021

School Mathematics Textbooks in China

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“…In addition to its importance in classical uniformization problems, this method has played a central role in recent developments on the uniformization of fractal metric spaces, cf. [1], [3], [4], [8], [17].…”

Section: The Corresponding Quotient Map Is πmentioning

confidence: 99%

Uniformization of planar domains by exhaustion

Rajala¹

2021

Preprint

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We study the method of finding conformal maps onto circle domains by approximating with finitely connected subdomains. Every domain D ⊂ Ĉ admits exhaustions, i.e., increasing sequences of finitely connected subdomains Dj whose union is D. By Koebe's theorem, each Dj admits a conformal map fD j from Dj onto a circle domain fD j (Dj ). Assuming fD j → f , our goal is to find out if f (D) is also a circle domain.We present a countably connected D with an exhaustion (Dj ) so that (fD j ) has a limit whose image is not a circle domain, and a domain Ω with an exhaustion (Ωj ) so that (fΩ j ) has a limit whose image has uncountably many non-point complementary components.On the other hand, we prove that every exhaustion (Dj) of a countably connected D admits a refinement so that the image of the corresponding limit map is a circle domain. Our result extends the He-Schramm theorem on the uniformization of countably connected domains and provides a new proof.

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Square Sierpiński carpets and Lattès maps

Cited by 6 publications

References 11 publications

Connectedness and local cut points of generalized Sierpinski carpets

Connectedness and local cut points of generalized Sierpinski carpets

Front Matter

Uniformization of planar domains by exhaustion

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