Mustafa SALTAN, Nisa ASLAN, Bünyamin DEMİR

A discrete chaotic dynamical system on the Sierpinski gasket

The Sierpinski gasket (also known as the Sierpinski triangle) is one of the fundamental models of self-similarsets. There have been many studies on different features of this set in the last decades. In this paper, initially we constructa dynamical system on the Sierpinski gasket by using expanding and folding maps. We then obtain a surprising shiftmap on the code set of the Sierpinski gasket, which represents the dynamical system, and we show that this dynamicalsystem is chaotic on the code set of the Sierpinski gasket with respect to the intrinsic metric. Finally, we provide analgorithm to compute periodic points for this dynamical system.

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[1] Alligood KT, Sauer TD, Yorke JA. Chaos: An Introduction to Dynamical Systems. New York, NY, USA: Springer- Verlag, 1996.
[2] Barnsley M. Fractals Everywhere. San Diego, CA, USA: Academic Press, 1988.
[3] Barnsley MF. Superfractals. New York, NY, USA: Cambridge University Press, 2006.
[4] Burago D, Burago Y, Ivanov S. A Course in Metric Geometry. Providence, RI, USA: AMS, 2001.
[5] Devaney RL. An Introduction to Chaotic Dynamical Systems. New York, NY, USA: Addison-Wesley Publishing Company, 1989.
[6] Ercai C. Chaos for the Sierpinski carpet. J Stat Phys 1997; 88: 979-984.
[7] Gulick D. Encounters with Chaos and Fractals. Boston, MA, USA: Academic Press, 1988.
[8] Hirsch MW, Smale S, Devaney RL. Differential Equations, Dynamical Systems, and an Introduction to Chaos. New York, NY, USA: Elsevier Academic Press, 2013.
[9] Huang W, Ye X. Devaney’s chaos or 2-scattering implies Li–Yorke’s chaos. Topology Appl 2002; 117: 259-272.
[10] Mandelbrot BB. The Fractal Geometry of Nature. San Francisco, CA, USA: W.H. Freeman and Company, 1982.
[11] Saltan M, Özdemir Y, Demir B. An explicit formula of the intrinsic metric on the Sierpinski gasket via code representation. Turk J Math 2018; 42: 716-725.
[12] Schroeder M. Fractals, Chaos, Power Laws. New York, NY, USA: W.H. Freeman, 1991.

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

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