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Abstract <p> A brief history of the Birkhoff curve and Wada basins is presented. The Birkhoff curves are found to be indecomposable continua that are the common boundary of two regions having a single composant. Therefore, a Birkhoff curve contains at most one fixed point. A simplest geometric model of the Birkhoff curve has been constructed by matching the tails of the composants of the indecomposable Knaster continuum having two composants. By analogy to Knaster’s continuum, examples of indecomposable continua having four and and six composants are constructed. By pairwise matching the tails of composants, the indecomposable continua are obtained that are common boundaries of three and four regions, respectively. There exist two and four topologically different matchings, respectively. Clearly, <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2675_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(2n\)</EquationSource> </InlineEquation>-composant indecomposable continuum admits <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2675_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(2^n\)</EquationSource> </InlineEquation> ways of matching. These geometric constructions demonstrate the anatomical structure of nonwandering continua possessing the Wada property for a dynamical system acting on the plane with a single hyperbolic fixed point. </p>

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On the geometric anatomy of a nonwandering continuum possessing the Wada property

  • 期刊论文

  • 发表日期：

    2025年8月23日

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  • D. W. Serow

摘要

Abstract

A brief history of the Birkhoff curve and Wada basins is presented. The Birkhoff curves are found to be indecomposable continua that are the common boundary of two regions having a single composant. Therefore, a Birkhoff curve contains at most one fixed point. A simplest geometric model of the Birkhoff curve has been constructed by matching the tails of the composants of the indecomposable Knaster continuum having two composants. By analogy to Knaster’s continuum, examples of indecomposable continua having four and and six composants are constructed. By pairwise matching the tails of composants, the indecomposable continua are obtained that are common boundaries of three and four regions, respectively. There exist two and four topologically different matchings, respectively. Clearly, \(2n\) -composant indecomposable continuum admits \(2^n\) ways of matching. These geometric constructions demonstrate the anatomical structure of nonwandering continua possessing the Wada property for a dynamical system acting on the plane with a single hyperbolic fixed point.