Bünyamin Demir

Dynamical systems have become a significant research area in recent years due to their rich structure and wide‐ranging applications. Analyzing their behavior often remains challenging, as even simple polynomial maps, such as the logistic map, can display highly complex dynamics. To circumvent this challenge, piecewise linear maps that are topologically conjugate to such polynomials are frequently considered. This article focuses on analyzing the dynamical behavior of a specific cubic polynomial. For this purpose, we first define a so‐called double‐sided tent map on a closed interval and represent it in base 2 to formulate its periodic points and investigate its dynamical properties. Furthermore, we construct a topological conjugacy map between the double‐sided tent map and the cubic polynomial, thereby demonstrating that the cubic polynomial also exhibits the same chaotic behavior as double‐sided tent map. We additionally prove that these dynamical systems possess periodic points of all even periods, but no periodic points of odd period except for a fixed point.

In this paper, we give a family of dynamical systems on the Cantor dust (Cx C) by using elements of 4th Dihedral group with shift map and then express them via code representations of the points of C x C. Moreover, we prove that these systems are chaotic in the sense of Devaney and also we investigate the topological equivalence of these dynamical systems.

The dynamical systems on the classical fractals can naturally be obtained with the help of their iterated function systems. In the recent years, different ways have been developed to define dynamical systems on the self similar sets. In this paper, we give composition functions by using expanding and folding mappings which generate the classical Sierpinski Gasket via the escape time algorithm. These functions also indicate dynamical systems on this fractal. We express the dynamical systems by using the code representations of the points. Then, we investigate whether these dynamical systems are topologically conjugate (equivalent) or not. Finally, we show that the dynamical systems are chaotic in the sense of Devaney and then we also compute and compare the periodic points.