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${\mathcal{M}}_{4}$ is regular-closed
Published online by Cambridge University Press: 10 April 2018
Abstract
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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- © Cambridge University Press, 2018
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