${\mathcal{M}}_{4}$ is regular-closed. (January 2020)

• Record Type: Journal Article
• Authors: HIMEKI, YUTARO; ISHII, YUTAKA
• Is Part Of: Ergodic theory and dynamical systems. Volume 40:Number 1(2020)
• Page Start: 213
• Page End: 220
• Publication Date: 2020-01
• Subjects: Ergodic theory -- Periodicals; Differentiable dynamical systems -- Periodicals; 515.42
• DOI: 10.1017/etds.2018.27
• Languages: English
• ISSNs: 0143-3857

Abstract

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 $n\geq 5$ .