Construction of a Dynamical System on the Sierpinski Gasket $SG(3)$

Year 2025, Volume: 18 Issue: 2, 415 - 424

Nisa Aslan , Mustafa Saltan

https://doi.org/10.36890/iejg.1689189

Abstract

Classical Sierpinski gasket is one of the fundamental examples of self-similar sets. This set is also named as mod-$2$ Sierpinski gasket. In this study, we mention the mod-$3$ Sierpinski gasket and obtain this self-similar set by using the escape time algorithm with some special mappings. Then, we express a dynamical system on the mod-$3$ Sierpinski gasket with the help of the code representations of its points.

Keywords

Mod-$3$ Sierpinski gasket , escape time algorithm , discrete dynamical system

Ethical Statement

No approval from the Board of Ethics is required.

Supporting Institution

Eskişehir Technical University

Project Number

22ADP340

Thanks

This work was supported by the Office of Scientific Research Projects Coordination at Eskişehir Technical University

References

• Aslan, N., Saltan, M., Demir, B.: The intrinsic metric formula and a chaotic dynamical system on the code set of the Sierpinski tetrahedron. Chaos Solitons Fractals 123, 422–428 (2019).
• Aslan, N.,Şeker, S., Saltan, M.: The investigation of chaos conditions of some dynamical systems on the Sierpinski propeller. Chaos Solitons Fractals 159, 112123 (2022).
• Barnsley, M.: Fractals Everywhere. Academic Press, Boston (1988).
• Cristea, L. L.: A geometric property of the Sierpinski carpet. Quaest. Math. 28(2), 251–262 (2005).
• Cristea, L. L., Steinsky, B.: Distances in Sierpinski graphs and on the Sierpinski gasket. Aequat. Math. 85, 201–219 (2013).
• Çınar, A. İ., Saltan, M.: On the computation of some code sets of the added Sierpinski triangle. Hacet. J. Math. Stat. 53(1), 130–144 (2024).
• Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications (2nd ed.). Wiley, Chichester (2003).
• Golmankhaneh, A. K.: Fractal Calculus and its Applications. World Scientific, Singapore (2022).
• Gu, J., Ye, Q., Xi, L.: Geodesics of higher-dimensional Sierpinski gasket. Fractals 27(1), 1950049 (2019).
• Gulick, D.: Encounters with Chaos and Fractals. Academic Press, San Diego (2012).
• Li, X., Liang, X., Xue, Y.: Geodesics of right isosceles Sierpi ´ nski gaskets and their relatives. Chaos Solitons Fractals 192, 115937 (2025).
• Mandelbrot, B. B.: The Fractal Geometry of Nature. Freeman, New York (1982).
• Özdemir, Y.: The intrinsic metric and geodesics on the Sierpinski gasket SG(3). Turk. J. Math. 43, 2741–2754 (2019).
• Özdemir, Y., Saltan, M., Demir, B.: The intrinsic metric on the box fractal. Bull. Iran. Math. Soc. 45(5), 1269–1281 (2019).
• Saltan, M., Aslan, N., Demir, B.: A discrete chaotic dynamical system on the Sierpinski gasket. Turk. J. Math. 43, 361–372 (2019).
• Seyhan, Ö., Aslan, N., Saltan, M.: Dynamical analysis of some special shift maps on discrete Sierpinski triangle. Chaos Solitons Fractals 178, 114382 (2024).
• Şen, A. İ., Saltan, M.: The formulization of the intrinsic metric on the added Sierpinski triangle by using the code representations. Turk. J. Math. 44(2), 356–377 (2020).
• Yılmaz, K., Özdemir, Y., Koparal, F. D.: Construction of new chaotic dynamical systems on a 2D Cantor set. Math. Commun. 28(2), 171–180 (2023).