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On connected components of fractal cubes. / Vaulin, Dmitrii Alekseevich; Drozdov, Dmitry Alekseevich; Tetenov, Andrei Viktorovich.

в: Trudy Instituta Matematiki i Mekhaniki UrO RAN, Том 26, № 2, 01.07.2020, стр. 98-107.

Результаты исследований: Научные публикации в периодических изданиях › статья › Рецензирование

Harvard

Vaulin, DA, Drozdov, DA & Tetenov, AV 2020, 'On connected components of fractal cubes', Trudy Instituta Matematiki i Mekhaniki UrO RAN, Том. 26, № 2, стр. 98-107. https://doi.org/10.21538/0134-4889-2020-26-2-98-107

APA

Vaulin, D. A., Drozdov, D. A., & Tetenov, A. V. (2020). On connected components of fractal cubes. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 26(2), 98-107. https://doi.org/10.21538/0134-4889-2020-26-2-98-107

Vancouver

Vaulin DA, Drozdov DA, Tetenov AV. On connected components of fractal cubes. Trudy Instituta Matematiki i Mekhaniki UrO RAN. 2020 июль 1;26(2):98-107. doi: 10.21538/0134-4889-2020-26-2-98-107

Author

Vaulin, Dmitrii Alekseevich ; Drozdov, Dmitry Alekseevich ; Tetenov, Andrei Viktorovich. / On connected components of fractal cubes. в: Trudy Instituta Matematiki i Mekhaniki UrO RAN. 2020 ; Том 26, № 2. стр. 98-107.

BibTeX

@article{9a0a2a177be847bdacf4532490ab8076,
title = "On connected components of fractal cubes",
abstract = "The paper shows an essential difference between fractal squares and fractal cubes. The topological classification of fractal squares proposed in 2013 by K.-S. Lau et al. was based on analyzing the properties of the Z2-periodic extension H = F + Z2of a fractal square F and of its complement Hc= R2\ H. A fractal square F ⊂ R2contains a connected component different from a line segment or a point if and only if the set Hccontains a bounded connected component. We show the existence of a fractal cube F in R3for which the set Hc= R3\H is connected whereas the set Q of connected components Kαof F possesses the following properties: Q is a totally disconnected self-similar subset of the hyperspace C(R3), it is bi-Lipschitz isomorphic to the Cantor set C1/5, all the sets Kα+Z3are connected and pairwise disjoint, and the Hausdorff dimensions dimH(Kα) of the components Kαassume all values from some closed interval [a, b].",
keywords = "Fractal cube, Fractal square, Hausdorff dimension, Hyperspace, Self-similar set, Superfractal, hyperspace, self-similar set, fractal cube, fractal square, CURVES, superfractal, INFINITE LENGTH",
author = "Vaulin, {Dmitrii Alekseevich} and Drozdov, {Dmitry Alekseevich} and Tetenov, {Andrei Viktorovich}",
note = "Ваулин Д.А., Дроздов Д.А., Тетенов А.В. О связных компонентах фрактальных кубов // Тр. Ин-та математики и механики УрО РАН. - 2020. - Т.26. - № 2. - С. 98-107",
year = "2020",
month = jul,
day = "1",
doi = "10.21538/0134-4889-2020-26-2-98-107",
language = "English",
volume = "26",
pages = "98--107",
journal = "Trudy Instituta Matematiki i Mekhaniki UrO RAN",
issn = "0134-4889",
publisher = "KRASOVSKII INST MATHEMATICS & MECHANICS URAL BRANCH RUSSIAN ACAD SCIENCES",
number = "2",

}

RIS

TY - JOUR

T1 - On connected components of fractal cubes

AU - Vaulin, Dmitrii Alekseevich

AU - Drozdov, Dmitry Alekseevich

AU - Tetenov, Andrei Viktorovich

N1 - Ваулин Д.А., Дроздов Д.А., Тетенов А.В. О связных компонентах фрактальных кубов // Тр. Ин-та математики и механики УрО РАН. - 2020. - Т.26. - № 2. - С. 98-107

PY - 2020/7/1

Y1 - 2020/7/1

N2 - The paper shows an essential difference between fractal squares and fractal cubes. The topological classification of fractal squares proposed in 2013 by K.-S. Lau et al. was based on analyzing the properties of the Z2-periodic extension H = F + Z2of a fractal square F and of its complement Hc= R2\ H. A fractal square F ⊂ R2contains a connected component different from a line segment or a point if and only if the set Hccontains a bounded connected component. We show the existence of a fractal cube F in R3for which the set Hc= R3\H is connected whereas the set Q of connected components Kαof F possesses the following properties: Q is a totally disconnected self-similar subset of the hyperspace C(R3), it is bi-Lipschitz isomorphic to the Cantor set C1/5, all the sets Kα+Z3are connected and pairwise disjoint, and the Hausdorff dimensions dimH(Kα) of the components Kαassume all values from some closed interval [a, b].

AB - The paper shows an essential difference between fractal squares and fractal cubes. The topological classification of fractal squares proposed in 2013 by K.-S. Lau et al. was based on analyzing the properties of the Z2-periodic extension H = F + Z2of a fractal square F and of its complement Hc= R2\ H. A fractal square F ⊂ R2contains a connected component different from a line segment or a point if and only if the set Hccontains a bounded connected component. We show the existence of a fractal cube F in R3for which the set Hc= R3\H is connected whereas the set Q of connected components Kαof F possesses the following properties: Q is a totally disconnected self-similar subset of the hyperspace C(R3), it is bi-Lipschitz isomorphic to the Cantor set C1/5, all the sets Kα+Z3are connected and pairwise disjoint, and the Hausdorff dimensions dimH(Kα) of the components Kαassume all values from some closed interval [a, b].

KW - Fractal cube

KW - Fractal square

KW - Hausdorff dimension

KW - Hyperspace

KW - Self-similar set

KW - Superfractal

KW - hyperspace

KW - self-similar set

KW - fractal cube

KW - fractal square

KW - CURVES

KW - superfractal

KW - INFINITE LENGTH

UR - http://www.scopus.com/inward/record.url?scp=85090531595&partnerID=8YFLogxK

UR - https://elibrary.ru/item.asp?id=42950651

U2 - 10.21538/0134-4889-2020-26-2-98-107

DO - 10.21538/0134-4889-2020-26-2-98-107

M3 - Article

AN - SCOPUS:85090531595

VL - 26

SP - 98

EP - 107

JO - Trudy Instituta Matematiki i Mekhaniki UrO RAN

JF - Trudy Instituta Matematiki i Mekhaniki UrO RAN

SN - 0134-4889

IS - 2

ER -

ID: 25310286