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

YUTARO HIMEKI, Yutaka Ishii

Research output: Contribution to journalArticle

1 Citation (Scopus)

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 languageEnglish
Pages (from-to)1-8
Number of pages8
JournalErgodic Theory and Dynamical Systems
DOIs
Publication statusAccepted/In press - Apr 10 2018

Fingerprint

Closed
Fractals
Contraction
Mandelbrot set
Self-similar Set
Geometric proof
Rotational symmetry
Iterated Function System
Locus
Fractal
Nonlinearity

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Cite this

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

In: Ergodic Theory and Dynamical Systems, 10.04.2018, p. 1-8.

Research output: Contribution to journalArticle

@article{fbf920859f184dd68291450768737518,
title = "{\mathcal{M}}:{4} is regular-closed",
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.).",
author = "YUTARO HIMEKI and Yutaka Ishii",
year = "2018",
month = "4",
day = "10",
doi = "10.1017/etds.2018.27",
language = "English",
pages = "1--8",
journal = "Ergodic Theory and Dynamical Systems",
issn = "0143-3857",
publisher = "Cambridge University Press",

}

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 -