Edgar, G. A.; Golds, Jeffrey A fractal dimension estimate for a graph-directed iterated function system of non-similarities. (English) Zbl 0944.28008 Indiana Univ. Math. J. 48, No. 2, 429-447 (1999). The authors investigate graph-directed iterated function systems determined by maps \(f_e\) having upper estimates of the form \(d(f_e(x),f_e(y))\leq r_ed(x,y)\) and prove that in this case the upper box counting dimension of the attractor is bounded above by the dimension associated to the Mauldin-Williams graph with ratios \(r_e\). They also show that if, in addition to the above upper estimate, the mappings \(f_e\) satisfy lower estimates of the form \(d(f_e(x),f_e(y))\geq r'_ed(x,y)\), then the Hausdorff dimension of the attractor is bounded below by the dimension associated to the Mauldin-Williams graph with ratios \(r'_e\) provided that the strong open set condition is valid. In the case \(r_e=r'_e\) the mappings are similarities and the consideration reduces to the dimension computation of R. D. Mauldin and S. C. Williams [Trans. Am. Math. Soc. 309, No. 2, 811-829 (1988; Zbl 0706.28007)]. Reviewer: Maarit Järvenpää (Jyväskylä) Cited in 14 Documents MSC: 28A80 Fractals 37F50 Small divisors, rotation domains and linearization in holomorphic dynamics 28A78 Hausdorff and packing measures Keywords:fractal dimension; graph-directed iterated function systems; IFS; upper box counting dimension; attractor; Mauldin-Williams graph; Hausdorff dimension Citations:Zbl 0706.28007 PDFBibTeX XMLCite \textit{G. A. Edgar} and \textit{J. Golds}, Indiana Univ. Math. J. 48, No. 2, 429--447 (1999; Zbl 0944.28008) Full Text: DOI arXiv Link Online Encyclopedia of Integer Sequences: Fixed point of the morphism 1 -> 12321, 2 -> 43234, 3 -> 21412, 4 -> 34143, starting from a(0) = 1.