Comparison of some dynamical systems on the quotient space of the Sierpinski tetrahedron

Year 2023, Volume: 72 Issue: 1, 229 - 239, 30.03.2023

Nisa Aslan Mustafa Saltan Bünyamin Demir

https://doi.org/10.31801/cfsuasmas.1126635

Cited By: 1

Abstract

In this paper, it is aimed to construct two different dynamical systems on the Sierpinski tetrahedron. To this end, we consider the dynamical systems on a quotient space of $\{ 0,1,2,3 \}^{\mathbb{N}}$ by using the code representations of the points on the Sierpinski tetrahedron. Finally, we compare the periodic points to investigate topological conjugacy of these dynamical systems and we conclude that they are not topologically equivalent.

Keywords

Sierpinski tetrahedron, quotient space, code representation, dynamical systems, topological conjugacy

References

• Alligood, K. T., Sauer, T. D., and Yorke, J. A., CHAOS: An Introduction to Dynamical Systems, Springer-Verlag, New York, 1996.
• Aslan, N., Saltan, M., and Demir, B., A different construction of the classical fractals via the escape time algorithm, Journal of Abstract and Computational Mathematics, 3(4) (2018), 1–15.
• Aslan, N., Saltan, M., and Demir, B., The intrinsic metric formula and a chaotic dynamical system on the code set of the Sierpinski tetrahedron, Chaos, Solitons and Fractals, 123 (2019), 422-428. Doi:10.1016/j.chaos.2019.04.018.
• Barnsley, M., Fractals Everywhere, 2nd ed. Academic Press, San Diego, 1988.
• Cristea, L. L., Steinsky B., Distances in Sierpinski graphs and on the Sierpinski gasket, Aequationes mathematicae, 85 (2013), 201-219. Doi: 10.1007/s00010-013-0197-7.
• Devaney, R. L., An introduction to Chaotic Dynamical Systems, Addison-Wesley Publishing Company, 1989.
• Devaney, R. L., Look D. M., Symbolic dynamics for a Sierpinski curve Julia set, J. Differ. Equ. Appl., 11(7) (2005), 581-596. Doi: 10.1080/10236190412331334473.
• Ercai, C., Chaos for the Sierpinski carpet, J. Stat. Phys., 88 (1997), 979-984.
• Gu, J., Ye, Q., and Xi, L., Geodesics of higher dimensional Sierpinski gasket, Fractals, 27(4) (2019), 1950049. Doi: 10.1142/S0218348X1950049X.
• Gulick, D., Encounters with Chaos and Fractals , Boston: Academic Press, 1988.
• Hirsch, M. W., Smale, S. and Devaney R. L., Differential Equations, Dynamical Systems, and an Introduction to Chaos, Elsevier Academic Press, 2013.
• Ozdemir, Y., Saltan, M. and Demir, B., The intrinsic metric on the box fractal, Bull. Iranian Math. Soc., 45(5) (2018), 1269-1281. Doi: 10.1007/s41980-018-00197-w.
• Peitgen, H. O., Jürgens, H. and Saupe, D., Chaos and Fractals, New Frontiers of Science, 2nd ed. Springer-Verlag, 2004.
• Saltan, M., Özdemir, Y., and Demir, B., Geodesic of the Sierpinski gasket, Fractals, 26(3) (2018), 1850024. Doi: 10.1142/S0218348X1850024X.
• Saltan, M., Özdemir, Y. and Demir, B., An explicit formula of the intrinsic metric on the Sierpinski gasket via code representation, Turkish J. Math., 42 (2018), 716-725. Doi:10.3906/mat-1702-55.
• Saltan, M., Intrinsic metrics on Sierpinski-like triangles and their geometric properties, Symmetry, 10(6) (2018), 204. Doi: 10.3390/sym10060204.
• Saltan, M., Aslan, N. and Demir, B., A discrete chaotic dynamical system on the Sierpinski gasket, Turkish J. Math., 43 (2019), 361-372. Doi: 10.3906/mat-1803-77.