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The Koch curve as a smooth manifold. (English) Zbl 1146.28300

Summary: We show that there exists a homeomorphism between the closed interval \([0,1]\subset \mathbb R\) and the Koch curve endowed with the subset topology of \(\mathbb R\). We use this homeomorphism to endow the Koch curve with the structure of a smooth manifold with boundary.

MSC:

28A80 Fractals
57R99 Differential topology
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References:

[1] Epstein, M.; Śniatycki, J., Fractal mechanics, Physica D, 220, 54-68 (2006) · Zbl 1098.74008
[2] Marshall, C. D., Calculus on subcartesian spaces, J Differential Geom, 10, 551-573 (1975) · Zbl 0317.58007
[3] von Koch, H., Une méthode géométrique élémentaire pour l’étude de certaines questions de la théorie des courbes planes, Acta Math, 30, 145-174 (1906) · JFM 37.0413.02
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