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The fractal “Frog”

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Abstract

In [1–3] some analytical properties were investigated of the Von Koch curve Γ θ , θ\((0,\tfrac{\pi } {4}) \) . In particular, it was shown that Γ θ is quasiconformal and not AC-removable. The natural question arises: Can one find a quasiconformal and not AC-removable curve essentially different from Γ θ in the sense that it is not diffeomorphic to Γ θ ? The present paper is an answer to the question. Namely, we construct a quasiconformal curve, calling the “Frog,” which is not AC-removable and not diffeomorphic to Γ θ for any θ\((0,\tfrac{\pi } {4}) \).

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References

  1. Ponomarev S., “AC-removability, Hausdorff dimension, and property (N),” Siberian Math. J., 35, No. 6, 1175–1183 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  2. Ponomarev S. P., “On Hausdorff dimensions of quasiconformal curves,” Siberian Math. J., 34, No. 4, 717–722 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  3. Ponomarev S. P., “Some properties of Von Koch’s curves,” Siberian Math. J., 48, No. 6, 1046–1059 (2007).

    Article  MathSciNet  Google Scholar 

  4. Edgar G., Measure, Topology, and Fractal Geometry, Springer-Verlag, New York, Berlin, and Heidelberg (1995).

    Google Scholar 

  5. Ahlfors L. V., Lectures on Quasiconformal Mappings, Van Nostrand Company, Princeton, N.J., Toronto, New York, and London (1966).

    MATH  Google Scholar 

  6. Rickman S., “Characterization of quasiconformal arcs,” Ann. Acad. Sci. Fenn. Ser. A I Math., 395, 7–30 (1966).

    MathSciNet  Google Scholar 

  7. Ghamsari M. and Herron D. A., “Higher dimensional Ahlfors regular sets and chordarc curves in ℝn,” Rocky Mountain J. Math., 28, No. 1, 191–222 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  8. Ghamsari M. and Herron D. A., “Bilipschitz homogeneous Jordan curves,” Trans. Amer. Math. Soc., 351, No. 8, 3197–3216 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  9. Dolzhenko E. P., “On ‘removing’ singularities of analytic functions,” Uspekhi Mat. Nauk, 18, No. 4, 135–141 (1963).

    MATH  Google Scholar 

  10. Hutchinson J., “Fractals and self-similarity,” Indiana Math. J., 30, No. 5, 713–747 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  11. Aseev V. V., Tetenov A. V., and Kravchenko A. S., “On selfsimilar Jordan curves on the plane,” Siberian Math. J., 44, No. 3, 379–386 (2003).

    Article  MathSciNet  Google Scholar 

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Correspondence to A. Gospodarczyk.

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Original Russian Text Copyright © 2012 Gospodarczyk A.

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 53, No. 4, pp. 794–804, July–August, 2012.

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Gospodarczyk, A. The fractal “Frog”. Sib Math J 53, 635–644 (2012). https://doi.org/10.1134/S0037446612040064

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  • DOI: https://doi.org/10.1134/S0037446612040064

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