M_4 is regular-closed. (January 2020) - HIMEKI, YUTARO; ISHII, YUTAKA
Abstract : For each n≥ 2, we investigate a family of iterated function systems which is parameterized by a common contraction ratio sD^× {sC:0<|s|<1} and possesses a rotational symmetry of order n . Let M_n be the locus of contraction ratio s for which the corresponding self-similar set is connected. The purpose of this paper is to show that M_n is regular-closed, that is, int M_n=M_n holds for n≥ 4 . This gives a new result for n=4 and a simple geometric proof of the previously known result by Bandt and Hung [Fractal n -gons and their Mandelbrot sets. Nonlinearity 21 (2008), 2653–2670] for n≥ 5 .