Analysis of some dynamical systems on the discrete vicsek fractal

In this thesis, the Discrete Vicsek Fractal (B^d), which is a totally disconnected fractal, is obtained as the attractor of an iterated function system, and code representations for its points are given. Using folding and expanding transformations on B^d, several dynamical systems are constructed, and their periodic points are studied. For one of these systems, a formula for the periodic points is found, and it is shown that the system is chaotic in the sense of Devaney. In the final parts of the thesis, a new family of dynamical systems is defined on B^d using maps from the symmetry group of the square together with the shift map. The periodic points of these systems are analyzed, and their chaotic behavior in the sense of Devaney is demonstrated. In addition, the topological equivalences between these transformations are also examined. The findings of this study provide a theoretical framework for understanding the relationship between fractal geometry and dynamic systems and contribute to a deeper comprehension of chaotic dynamics on fractals. Moreover, the analysis conducted on entirely disconnected fractals represents an innovative approach with potential applications in both mathematics and engineering.