Euclidean realizations of Mauldin-Williams graphs

DENİZ A., Kocak S., Ozdemir Y., ÜREYEN A. E.

JOURNAL OF GEOMETRY, cilt.104, sa.2, ss.257-263, 2013 (ESCI) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 104 Sayı: 2
  • Basım Tarihi: 2013
  • Doi Numarası: 10.1007/s00022-013-0164-4
  • Dergi Adı: JOURNAL OF GEOMETRY
  • Derginin Tarandığı İndeksler: Emerging Sources Citation Index (ESCI), Scopus
  • Sayfa Sayıları: ss.257-263
  • Anahtar Kelimeler: Graph-directed fractals, iterated function systems, weighted graphs, open set condition
  • Anadolu Üniversitesi Adresli: Evet

Özet

"Graph-directed" fractals are collections of metric spaces, each of which can be expressed as a union of several scaled copies of spaces from the collection. They give rise to weighted, directed graphs where the term comes from. We show in this note that any (finite) weighted, directed graph (with weights between 0 and 1) can be realized in a Euclidean space in the sense that, starting from the graph one can define a system of similitudes (with the similarity ratios being the given weights) on an appropriate Euclidean space. The point is that these maps satisfy a certain property (called the open set condition) so that the theory of Mauldin-Williams can be applied to compute the dimension of the emerging fractals. Additionally, we give a novel example of a system of graph-directed fractals.