On connected components of fractal cubes

Dmitrii Alekseevich Vaulin, Dmitry Alekseevich Drozdov, Andrei Viktorovich Tetenov

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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].

Translated title of the contributionО связных компонентах фрактальных кубов
Original languageEnglish
Pages (from-to)98-107
Number of pages10
JournalTrudy Instituta Matematiki i Mekhaniki UrO RAN
Issue number2
Publication statusPublished - 1 Jul 2020


  • Fractal cube
  • Fractal square
  • Hausdorff dimension
  • Hyperspace
  • Self-similar set
  • Superfractal
  • hyperspace
  • self-similar set
  • fractal cube
  • fractal square
  • superfractal

State classification of scientific and technological information



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