Fractal squares with finitely many connected components. (18th February 2021)

• Record Type: Journal Article
• Authors: Xiao, Jian-Ci
• Is Part Of: Nonlinearity. Volume 34:Number 4(2021)
• Page Start: 1817
• Page End: 1836
• Publication Date: 2021-02-18
• Subjects: fractal square -- connected component -- graph; Primary 28A80; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/abd611
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: In this paper, we present an effective method to characterize completely when a disconnected fractal square has only finitely many connected components. Our method is to establish some graph structures on fractal squares to reveal the evolution of the connectedness during their geometric iterated construction. We also prove that every fractal square contains either finitely or uncountably many connected components. A few examples, including the construction of fractal squares with exactly m ⩾ 2 connected components, are added in addition.