Abstract
We study 4 × 4-labyrinth fractals, which are self similar dendrites. For all 4 × 4-labyrinth fractals we answer the question, whether there is a curve of finite length in the fractal from one point to another point in the fractal. In the first case, between any two points in the fractal there is a unique arc a, the length of a is infinite, and the set of points, where no tangent exists to a, is dense in a. In the second case, there are also pairs of points between that there is a unique arc of finite length.
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Allouche J.-P., Shallit J.: Automatic Sequences: Theory, Applications, Generalizations, pp. 155–156. Cambridge University Press, Cambridge England (2003)
Cristea, L.L.: On the connectedness of limit net sets. Submitted (2008)
Falconer K.J.: Fractal Geometry, Mathematical Foundations and Applications. John & Wiley Sons, Chichester (1990)
Hilbert D.: Über die stetige Abbildung einer Linie auf ein Flachenstück. Math. Ann. 38, 459–460 (1891)
Kuratowski K.: Topology. vol. 2. Academic Press, New York and London (1968)
Moran P.A.P.: Additive functions of intervals and Hausdorff measure. Proc. Cambridge Philos. Soc. 42, 15–23 (1946)
Peano G.: Sur une courbe, qui remplit une aire plane. Math. Ann. 36, 157–160 (1890)
Pesin Y., Weiss H.: On the dimension of deterministic and random cantor-like sets. Comm. Math. Phys. 182, 105–153 (1996)
Pesin Y., Weiss H.: A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions. J. Stat. Phys. 86, 233–275 (1997)
Seneta, E.: Non-negative matrices and Markov chains, 2nd edn. Springer Verlag, New York (1973, 1981, 2006)
Tricot C.: Curves and Fractal Dimension. Springer-Verlag, Paris (1993)
von Koch H.: Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire. Archiv för Matemat. 1, 681–702 (1904)
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)
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The authors are supported by the Austrian Science Fund (FWF), Project P-20412.
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Cristea, L.L., Steinsky, B. Curves of infinite length in 4 × 4-labyrinth fractals. Geom Dedicata 141, 1–17 (2009). https://doi.org/10.1007/s10711-008-9340-3
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DOI: https://doi.org/10.1007/s10711-008-9340-3