On the computation of some code sets of the added Sierpinski triangle

Abstract

In recent years, the intrinsic metrics have been formulated on the classical fractals. In particular, Sierpinski-like triangles such as equilateral, isosceles, scalene, added and mod$-3$ Sierpinski triangle have been considered in many different studies. The intrinsic metrics can be defined in different ways. One of the methods applied to obtain the intrinsic metric formulas is to use the code representations of the points on these self-similar sets. To define the intrinsic metrics via the code representations of the points on fractals makes also possible to investigate different geometrical, topological properties and geodesics of these sets. In this paper, we investigate some circles and closed sets of the added Sierpinski triangle and express them as the code sets by using its intrinsic metric.

Keywords

• Sierpinski gasket
• fractal
• intrinsic metric
• code representation

Supporting Institution

TÜBİTAK

Project Number

118F356

Thanks

The authors would like to thank TÜBİTAK for their support of the project.

References

1. [1] N. Aslan, M. Saltan and B. Demir, The intrinsic metric formula and a chaotic dynamical system on the code set of the Sierpinski tetrahedron, Chaos Soliton Fract. 123, 422-428, 2019.
2. [2] N. Aslan, M. Saltan and B. Demir, On Topological conjugacy of some chaotic dynamical systems on the Sierpinski gasket, Filomat 35 (7), 2317–2391, 2021.
3. [3] N. Aslan and M. Saltan, On the construction of chaotic dynamical systems on the box fractal, Researches in Mathematics 29 (2) 3–14, 2021.
4. [4] N. Aslan, S. Şeker and M. Saltan, The investigation of chaos conditions of some dynamical systems on the Sierpinski propeller, Chaos Soliton Fract. 159, 112123, 2022.
5. [5] N. Aslan and İ. Aslan, Approximation to the classical fractals by using non-affine contraction mappings, Port. Math. 79, 45–60, 2022.
6. [6] M. Barnsley, Fractals Everywhere, Academic Press, San Diego, CA, USA, 1988.
7. [7] L.L. Cristea and B. Steinsky, Distances in Sierpinski graphs and on the Sierpinski gasket, Aequationes Math. 85, 201-219, 2013.
8. [8] M. Denker and H. Sato, Sierpinski gasket as a Martin boundary II (the intrinsic metric), Publ. Res. Inst. Math. Sci. 35, 769-794, 1999.

9. [9] G. Edgar, Measure, Topology, and Fractal Geometry, Springer, New York, NY, USA, 2008.
10. [10] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley, Hoboken, NJ, USA, 2004.
11. [11] P. Grabner and R.F. Tichy, Equidistribution and Brownian motion on the Sierpinski gasket, Monatshefte für Mathematik 125, 147-164, 1998.
12. [12] A.M. Hinz and A. Schief, The average distance on the Sierpinski gasket, Probab. Theory Relat. Fields 87, 129–138, 1990.
13. [13] D. Romik, Shortest paths in the Tower of Hanoi graph and finite automata, SIAM J. Discret. Math. 20, 610–622, 2006.
14. [14] M. Saltan, Y. Özdemir and B. Demir, An explicit formula of the intrinsic metric on the Sierpinski gasket via code representation, Turk. J. Math. 42, 716–725, 2018.
15. [15] M. Saltan, Y. Özdemir and B. Demir, Geodesics of the Sierpinski gasket, Fractals 26, 1850024, 2018.
16. [16] M. Saltan, Some interesting code sets of the Sierpinski triangle equipped with the intrinsic metric, IJAMAS 57, 152-160, 2018.
17. [17] M. Saltan, Intrinsic metrics on Sierpinski-like triangles and their geometric properties, Symmetry 10, 204, DOI:10.3390/sym10060204, 2018.
18. [18] M. Saltan, N. Aslan and B. Demir, A discrete chaotic dynamical system on the Sierpinski gasket, Turk. J. Math. 43, 361-372, 2019.
19. [19] R.S. Strichartz, Isoperimetric estimates on Sierpinski gasket type fractals, Trans. Am. Math. Soc. 351, 1705-1752, 1999.
20. [20] A.İ. Şen and M. Saltan, The formulization of the intrinsic metric on the added Sierpinski triangle by using the code representations, Turk. J. Math. 42, 716-725, 2018.