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${\mathcal{M}}_{4}$ is regular-closed



For each $n\geq 2$ , we investigate a family of iterated function systems which is parameterized by a common contraction ratio $s\in \mathbb{D}^{\times }\equiv \{s\in \mathbb{C}:0<|s|<1\}$ and possesses a rotational symmetry of order $n$ . Let ${\mathcal{M}}_{n}$ 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 ${\mathcal{M}}_{n}$ is regular-closed, that is, $\overline{\text{int}\,{\mathcal{M}}_{n}}={\mathcal{M}}_{n}$ holds for $n\geq 4$ . 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 $n\geq 5$ .



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[BanHu] Bandt, C. and Hung, N. V.. Fractal n-gons and their Mandelbrot sets. Nonlinearity 21(11) (2008), 26532670.
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