Euclidean realizations of Mauldin-Williams graphs

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

JOURNAL OF GEOMETRY, vol.104, no.2, pp.257-263, 2013 (ESCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 104 Issue: 2
  • Publication Date: 2013
  • Doi Number: 10.1007/s00022-013-0164-4
  • Journal Name: JOURNAL OF GEOMETRY
  • Journal Indexes: Emerging Sources Citation Index (ESCI), Scopus
  • Page Numbers: pp.257-263
  • Keywords: Graph-directed fractals, iterated function systems, weighted graphs, open set condition
  • Anadolu University Affiliated: Yes

Abstract

"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.