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 $\mathbb{Z}^2$-periodic extension $H=F+\mathbb{Z}^2$ of a fractal square $F$ and of its complement $H^c=\mathbb{R}^2\setminus H$. A fractal square $F\subset\mathbb{R}^2$ contains a connected component different from a line segment or a point if and only if the set $H^c$ contains a bounded connected component. We show the existence of a fractal cube $F$ in $\mathbb R^3$ for which the set $H^c=\mathbb{R}^3\setminus H$ is connected whereas the set $Q$ of connected components $K_\alpha$ of $F$ possesses the following properties: $Q$ is a totally disconnected self-similar subset of the hyperspace $C(\mathbb R^3)$, it is bi-Lipschitz isomorphic to the Cantor set $C_{1/5}$, all the sets $K_\alpha+\mathbb{Z}^3$ are connected and pairwise disjoint, and the Hausdorff dimensions $\dim_H(K_\alpha)$ of the components $K_\alpha$ assume all values from some closed interval $[a,b]$.
Keywords: fractal square, fractal cube, superfractal, self-similar set, hyperspace, Hausdorff dimension
Received April 6, 2020
Revised April 20, 2020
Accepted May 11, 2020
Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 18-01-00420).
Dmitrii Alekseevich Vaulin, Gorno-Altaisk State University, Gorno-Altaisk, 649000 Russia, e-mail: d_warrant@mail.ru
Dmitry Alekseevich Drozdov, Gorno-Altaisk State University, Gorno-Altaisk, 649000 Russia, e-mail: dimalek97@yandex.ru
Andrei Viktorovich Tetenov, Dr. Phys.-Math. Sci., Prof., Gorno-Altaisk State University, Gorno-Altaisk, 649000 Russia, Novosibirsk State University, Novosibirsk, 630090 Russia, e-mail: atet@mail.ru
REFERENCES
1. Barnsley M.F., Hutchinson J.E., Stenflo O. V -variable fractals: Fractals with partial self similarity. Adv. Math., 2008, vol. 218, no. 6, pp. 2051–2088. doi: 10.1016/j.aim.2008.04.011
2. Bonk M., Merenkov S. Quasisymmetric rigidity of square Sierpinski carpets. Annals Math., 2013, vol. 177, no. 2, pp. 591–643. doi: 10.4007/annals.2013.177.2.5
3. Cristea L.L., Steinsky B. Curves of infinite length in 4×4-labyrinth fractals. Geom. Dedicata, 2009, vol. 141, pp. 1–17. doi: 10.1007/s10711-008-9340-3
4. Cristea L.L., Steinsky B. Curves of infinite length in labyrinth fractals. Proc. Edinb. Math. Soc., II. Ser., 2011, vol. 54, no. 2, pp. 329–344. doi: 10.1017/S0013091509000169
5. Falconer K.J. Fractal geometry: mathematical foundations and applications. N Y: J. Wiley and Sons, 1990, 288 p. ISBN: 0471922870 .
6. Hutchinson J. Fractals and self-similarity. Indiana Univ. Math. J., 1981, vol. 30, no. 5, pp. 713–747. DOI: 10.1512/iumj.1981.30.30055
7. Lau K.S., Luo J.J., Rao H. Topological structure of fractal squares. Math. Proc. Camb. Phil. Soc., 2013, vol. 155, no. 1, pp. 73–86. doi: 10.1017/S0305004112000692
8. Luo J.J., Liu J.C. On the classification of fractal squares. Fractals, 2016, vol. 24, no. 1, art.-no. 1650008. doi: 10.1142/S0218348X16500080
9. Ruan H.J., Wang Y. Topological invariants and Lipschitz equivalence of fractal squares. J. Math. Anal. Appl., 2017, vol. 451, no. 1, pp. 327–344. doi: 10.1016/j.jmaa.2017.02.012
10. Tetenov A.V. Finiteness properties for self-similar sets. arXiv:2003.04202 [math.MG]
11. Tetenov A.V., Drozdov D.A. On the connected components of fractal cubes. arXiv:2002.02920 [math.MG]. 6 p.
Cite this article as: D.A. Vaulin, D.A. Drozdov, A.V. Tetenov. On connected components of fractal cubes. Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 2, pp. 98–107.
