Abstract
In 2008, the author introduced a class of space-filling curves associated to fractals that satisfy the a special property. These structures admit geodesic laminations on the disc, which help to understand the geometrical and the dynamical properties of the space-filling curves. In the present article we study the relation between the symmetries of the laminations and the fractals. In particular we prove that the group of symmetries of the lamination is isomorphic to a subgroup of the full group of symmetries of the fractal. We extend the results to a larger class of fractals using the concept of sub-IFS.
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Akiyama S., Loridant B.: Boundary parametrization of self-affine tiles. J. Math. Soc. Jpn. 63, 525–579 (2011)
Barnsley M.F.: Fractals Everywhere, 2nd edn. Academic, Boston (1993)
Casson A., Bleiler S.: Automorphisms of Surfaces after Nielsen and Thurston. Cambridge University, Cambridge (1988)
Falconer K.: Fractal geometry: mathematical foundations and applications. Wiley, Chichester (1990)
Falconer K., O’Connor J.: Symmetry and enumeration of self-similar fractals. Bull. London Math. Soc. 39, 272–282 (2007)
Hata M.: On the structure of self similar sets. Jpn. J. Appl. Math. 2, 381–414 (1985)
Hilbert, D.: Über die stetige Abbildung einer Linie auf ein Flächenstück, Mathematische Annalen 38 (1891), 459–460. Gesammelte Abhandlungen Vol III, 1–2, Springer-Verlag, Berlin, (1935)
Keller, K.: Invariant factors, Julia equivalences and the (abstract) Mandelbrot set, Lecture Notes in Mathematics, 1732, Springer, Berlin, (2000)
Knopp K.: Einheitliche Erzeugung und Darstellung der Kurven von Peano, Osgood und von Koch. Arch. Math. Phys. 26, 103–115 (1917)
Lebesgue H.: Leçons sur l’Intégration et la Recherche des Fonctions Primitives. Gauthiers-Villars, Paris (1904)
Peano G.: Sur une courbe qui remplit toute une aire plane. Math. Ann. 36, 157–160 (1890)
Peano, G.: La Curva di Peano Nel: Formulario Mathematico, vol 5, Frates Bocca, Turin 239–240 (1908)
Sagan H.: Space-Filling curves. Springer-Verlag, New York (1994)
Sierpiński, W.: Sur une nouvelle courbe continue qui remplit toute une aire plane, Bull. Acad. Sci. de Cracovie, Série A, 462–478 (1912)
Sirvent V.F.: Geodesic laminations as geometric realizations of Pisot substitutions. Ergodic Theory Dynam. Syst. 20, 1253–1266 (2000)
Sirvent, V.F.: Hilbert’s space filling curves and geodesic laminations. Math. Phys. Electron. J. 9, (2003)
Sirvent V.F.: Geodesic laminations as geometric realizations of Arnoux-Rauzy sequences. Bull. Belg. Math. Soc. Simon Stevin 10, 221–229 (2003)
Sirvent V.F.: Space filling curves and geodesic laminations. Geometriae Dedicata. 135, 1–14 (2008)
Thurston W.P.: On the combinatorics and dynamics of iterated rational maps. Preprint, Princeton (1985)
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Communicated by K. Schmidt.
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Sirvent, V.F. Space-filling curves and geodesic laminations. II: Symmetries. Monatsh Math 166, 543–558 (2012). https://doi.org/10.1007/s00605-012-0406-9
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DOI: https://doi.org/10.1007/s00605-012-0406-9