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Rigidity Theorem for Self-Affine Arcs

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Abstract

It has been known for more than a decade that, if a self-similar arc \(\gamma \) can be shifted along itself by similarity maps that are arbitrarily close to identity, then \(\gamma \) is a straight line segment. We extend this statement to the class of self-affine arcs and prove that each self-affine arc admitting affine shifts that may be arbitrarily close to identity is a segment of a parabola or a straight line.

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Funding

This work was supported by the Mathematical Center in Akademgorodok with the Ministry of Science and Higher Education of the Russian Federation, agreement no. 075-15-2019-1613.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, 630090, Novosibirsk, Russia

    A. V. Tetenov

  2. Novosibirsk State University, 630090, Novosibirsk, Russia

    A. V. Tetenov & O. A. Chelkanova

  3. Gorno-Altaisk State University, 649000, Gorno-Altaisk, Russia

    A. V. Tetenov

Authors
  1. A. V. Tetenov
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  2. O. A. Chelkanova
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Correspondence to A. V. Tetenov.

Additional information

Translated by I. Ruzanova

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Tetenov, A.V., Chelkanova, O.A. Rigidity Theorem for Self-Affine Arcs. Dokl. Math. 103, 81–84 (2021). https://doi.org/10.1134/S1064562421020058

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  • DOI: https://doi.org/10.1134/S1064562421020058

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