Mandelbrot set for fractal n-gons and zeros of power series

Abstract

We give a framework to study the connectedness of the set of zeros of power series with coefficients in a finite subset G⊂C. We prove that the set of zeros in the unit disk is connected and locally connected if some graph on the set G of coefficients is connected. Furthermore, we apply this result to the study of the Mandelbrot set Mn for fractal n-gons. We prove that Mn is connected and locally connected for any n.