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

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Title: On the geometric anatomy of a nonwandering continuum possessing the Wada property.
Authors: Serow, D. W.1 (AUTHOR) dimusum@yandex.ru
Source: Theoretical & Mathematical Physics. Aug2025, Vol. 224 Issue 2, p1428-1436. 9p.
Subjects: Dynamical systems, Mathematical continuum, Topological property, Geometric modeling
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, -composant indecomposable continuum admits 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. [ABSTRACT FROM AUTHOR]
Copyright of Theoretical & Mathematical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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An: 187497960
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  Data: 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, -composant indecomposable continuum admits 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. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Theoretical & Mathematical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1134/S0040577925080070
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        Text: English
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      – SubjectFull: Mathematical continuum
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      – SubjectFull: Topological property
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      – SubjectFull: Geometric modeling
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