A discrete chaotic dynamical system on the Sierpinski gasket

Mustafa Saltan,Nisa Aslan,Bünyamin Demi̇r

doi:10.3906/mat-1803-77

Abstract

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

Highlights

Fractals were introduced as the geometry of nature by Mandelbrot [10] and studied in various fields including mathematics, social science, computer science, engineering, economics, physics, chemistry, and biology
We construct a discrete dynamical system {S; F } by using the code representations of the points on the Sierpinski gasket and we show that this dynamical system is chaotic in the sense of Devaney with respect to the intrinsic metric d in Proposition 2.2
As seen in the example of the right Sierpinski gasket in (1.1), a dynamical system can be naturally defined on any fractal that can be expressed by the attractor of an iterated function system (IFS)

Summary

Introduction

Fractals were introduced as the geometry of nature by Mandelbrot [10] and studied in various fields including mathematics, social science, computer science, engineering, economics, physics, chemistry, and biology (for details see [1,2,3, 5, 7, 8, 12]). We construct a discrete dynamical system {S; F } by using the code representations of the points on the Sierpinski gasket and we show that this dynamical system is chaotic in the sense of Devaney with respect to the intrinsic metric d in Proposition 2.2. To this end, we first obtain a dynamical system on S by using expanding and folding maps. In [11], an explicit formula that gives the intrinsic distance between any two points of the code set of S is defined as follows: Definition 1.1 Let a1a2 .

We also denote the isosceles trapezoid

It is clear that

Conclusion