Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-04T18:41:34.824Z Has data issue: false hasContentIssue false
  • English
  • Français

Curves of infinite length in labyrinth fractals

Part of: General convexity Real and complex geometry Computational aspects in algebraic geometry Functions of several variables Classical measure theory

Published online by Cambridge University Press:  30 March 2011

Ligia L. Cristea  and
Bertran Steinsky
Ligia L. Cristea
Affiliation:
Technical University of Graz, Steyrergasse 30, 8010 Graz, Austria ([email protected])
Bertran Steinsky
Affiliation:
Department of Knowledge-Based Mathematical Systems, Johannes Kepler University Linz, Altenberger Strasse 69, 4040 Linz, Austria ([email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We define an infinite class of fractals, called horizontally and vertically blocked labyrinth fractals, which are dendrites and special Sierpiński carpets. Between any two points in the fractal there is a unique arc α; the length of α is infinite and the set of points where no tangent to α exists is dense in α.

Keywords

infinite curve lengthfractaldendritetreeinfinite distance

MSC classification

Secondary: 14Q05: Curves 26B05: Continuity and differentiation questions 28A75: Length, area, volume, other geometric measure theory 28A80: Fractals 51M25: Length, area and volume 52A38: Length, area, volume
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 2 , June 2011 , pp. 329 - 344
Copyright
Copyright © Edinburgh Mathematical Society 2011

References

1.Akiyama, S. and Thuswaldner, J. M., A survey on topological properties of tiles related to number systems, Geom. Dedicata 109 (2004), 89–105.Google Scholar
2.Allouche, J.-P. and Shallit, J., Automatic sequences: theory, applications, generalizations, pp. 155–156 (Cambridge University Press, 2003).Google Scholar
3.Bandt, C. and Keller, K., Self-similar sets, II, A simple approach to the topological structure of fractals, Math. Nachr. 154 (1991), 27–39.CrossRefGoogle Scholar
4.Cristea, L. L., On the connectedness of limit net sets, Topol. Applic. 155(16) (2008), 1808–1819.CrossRefGoogle Scholar
5.Cristea, L. L. and Steinsky, B., Curves of infinite length in 4 × 4 labyrinth fractals, Geom. Dedicata 141 (2009), 1–17.CrossRefGoogle Scholar
6.Falconer, K. J., Fractal geometry, mathematical foundations and applications (Wiley, 1990).Google Scholar
7.Hata, M., On the structure of self-similar sets, Jpn. J. Appl. Math. 2(2) (1985), 381–414.CrossRefGoogle Scholar
8.Hilbert, D., Über die stetige Abbildung einer Linie auf ein Flächenstück, Math. Annalen 38 (1891), 459–460.Google Scholar
9.Kuratowski, K., Topology, Volume 2 (Academic Press, New York, 1968).Google Scholar
10.Moran, P. A. P., Additive functions of intervals and Hausdorff measure, Proc. Camb. Phil. Soc. 42 (1946), 15–23.Google Scholar
11.Peano, G., Sur une courbe, qui remplit une aire plane, Math. Annalen 36 (1890), 157–160.CrossRefGoogle Scholar
12.Pesin, Y. and Weiss, H., On the dimension of deterministic and random Cantor-like sets, Commun. Math. Phys. 182 (1996), 105–153.CrossRefGoogle Scholar
13.Pesin, Y. and Weiss, H., A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions, J. Statist. Phys. 86 (1997), 233–275.CrossRefGoogle Scholar
14.Seneta, E., Non-negative matrices and Markov chains, 2nd edn (Springer, 2006).Google Scholar
15.Tricot, C., Curves and fractal dimension (Springer, 1993).Google Scholar
16.von Koch, H., Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire, Ark. Mat. 1 (1904), 681–702.Google Scholar
17.von Koch, H., Une méthode géométrique élémentaire pour l'étude de certaines questions de la théorie des courbes planes, Acta Math. 30 (1906), 145–174.CrossRefGoogle Scholar