### Abstract

For each (Formula presented.), we investigate a family of iterated function systems which is parameterized by a common contraction ratio (Formula presented.) and possesses a rotational symmetry of order (Formula presented.). Let (Formula presented.) be the locus of contraction ratio (Formula presented.) for which the corresponding self-similar set is connected. The purpose of this paper is to show that (Formula presented.) is regular-closed, that is, (Formula presented.) holds for (Formula presented.). This gives a new result for (Formula presented.) and a simple geometric proof of the previously known result by Bandt and Hung [Fractal (Formula presented.)-gons and their Mandelbrot sets. Nonlinearity 21 (2008), 2653–2670] for (Formula presented.).

Original language | English |
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Pages (from-to) | 1-8 |

Number of pages | 8 |

Journal | Ergodic Theory and Dynamical Systems |

DOIs | |

Publication status | Accepted/In press - Apr 10 2018 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Ergodic Theory and Dynamical Systems*, 1-8. https://doi.org/10.1017/etds.2018.27

**{\mathcal{M}}:{4} is regular-closed.** / HIMEKI, YUTARO; Ishii, Yutaka.

Research output: Contribution to journal › Article

*Ergodic Theory and Dynamical Systems*, pp. 1-8. https://doi.org/10.1017/etds.2018.27

}

TY - JOUR

T1 - {\mathcal{M}}:{4} is regular-closed

AU - HIMEKI, YUTARO

AU - Ishii, Yutaka

PY - 2018/4/10

Y1 - 2018/4/10

N2 - For each (Formula presented.), we investigate a family of iterated function systems which is parameterized by a common contraction ratio (Formula presented.) and possesses a rotational symmetry of order (Formula presented.). Let (Formula presented.) be the locus of contraction ratio (Formula presented.) for which the corresponding self-similar set is connected. The purpose of this paper is to show that (Formula presented.) is regular-closed, that is, (Formula presented.) holds for (Formula presented.). This gives a new result for (Formula presented.) and a simple geometric proof of the previously known result by Bandt and Hung [Fractal (Formula presented.)-gons and their Mandelbrot sets. Nonlinearity 21 (2008), 2653–2670] for (Formula presented.).

AB - For each (Formula presented.), we investigate a family of iterated function systems which is parameterized by a common contraction ratio (Formula presented.) and possesses a rotational symmetry of order (Formula presented.). Let (Formula presented.) be the locus of contraction ratio (Formula presented.) for which the corresponding self-similar set is connected. The purpose of this paper is to show that (Formula presented.) is regular-closed, that is, (Formula presented.) holds for (Formula presented.). This gives a new result for (Formula presented.) and a simple geometric proof of the previously known result by Bandt and Hung [Fractal (Formula presented.)-gons and their Mandelbrot sets. Nonlinearity 21 (2008), 2653–2670] for (Formula presented.).

UR - http://www.scopus.com/inward/record.url?scp=85045133702&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85045133702&partnerID=8YFLogxK

U2 - 10.1017/etds.2018.27

DO - 10.1017/etds.2018.27

M3 - Article

AN - SCOPUS:85045133702

SP - 1

EP - 8

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

ER -