Fractals via Self-Similar Group of Fisher Contractions

The fractal notion was unveiled by Mandelbrot in 1975, and a number of researchers continued to popularize the fractal theory. The chaotic structures that are frequently encountered in various complex systems can be addressed by the mathematical features of fractals. The iterated function system is constructed as an application of the theory of discrete dynamical systems for creating fractals and other similar sets with respect to the iterative algebraic condition. In this context, self-similar and strong self-similar groups of Fisher contractions are defined and explored. These groups are generated as the invariant set of a strong self-similar group of Fisher contractions. This study constructs a group of strong self-similar Fisher contractions by using the idea of Hutchinson-Barnsley idea in Fisher iterated function system. The strong self-similarity of a compact topological group can be used to generate the fractal attractor. Finally, we explore the concrete connection between the profinite group of Fisher contractions and the strong type of self-similar group consists of Fisher contractions.