Zeros of some random polynomials and connectedness locus of family of fractals (Research on the Theory of Random Dynamical Systems and Fractal Geometry)

Abstract

We consider the closure of the set of zeros of polynomials with complex coefficients randomly chosen from a "good" compact subset of complex plane. We prove that the closure of the set of zeros is connected. Furthermore, we apply this result to the study of connectedness locus (Mn) of fractal n-gons and the remarkable subset M^0n. Fracal n-gons and Mn are introduced by C. Bandt and N. V. Hung (2008). It is already known that M₂, M^0₂ and M₃ are connected (T. Bousch (1988), Y. Himeki under supervision Y. Ishii(2018)). We prove that for each n = 2, 3, 4, 5, 6, Mn is connected and for all n M^0n is connected.