MMN-3199

# Fixed point theorem and self-similarity on mixed Vicsek patterns

**Levente Simon**, Babes-Bolyai University, Faculty of Mathematics and Computer Science, 1 M. Kogalniceanu St., 400084 Cluj-Napoca, Romania,`simonl@math.ubbcluj.ro`

## Abstract

This paper highlights a fixed point result on sets constructed by finite sequence patterns. Using a fractal operator, a mixed pattern generated by the a finite sequence patterns constructs a set. We also define a distance based the areas of the black squares and we get that this constructs a complete metric space on the mixed pattern's set.
As a highlighted result, we show that the fractal operator has an unique fixed point of the set generated by the mixed pattern. We apply the result on Vicsek fractals such that we get that the results also hold for mixed Vicsek patterns.
Using circle patterns, we also interpret the result on on sets mixed patterns based on touching circles. We accent a fixed point theorem on the sets mixed patterns built by iterated function systems and the distances calculated between the areas these geometric shapes.

Vol. 22 (2021), No. 2, pp. 977-989

DOI: 10.18514/MMN.2021.3199