A Fractal Dimension Estimate for a Graph-directed Iterated Function System of Non-similarities

Suppose a graph-directed iterated function system consists of maps fe with upper estimates of the form d ( fe(x),fe(y) ) ≤ red(x,y). Then the fractal dimension of the attractor Kv of the IFS is bounded above by the dimension associated to the Mauldin–Williams graph with ratios re. Suppose the maps fe also have lower estimates of the form d ( fe(x),fe(y) ) ≥ r′ ed(x,y) and that the IFS also satisfies the strong open set condition. Then the fractal dimension of the attractor Kv of the IFS is bounded below by the dimension associated to the Mauldin–Williams graph with ratios r′ e. When re = r ′ e, then the maps are similarities and this reduces to the dimension computation of Mauldin and Williams for that case. 0. Introduction. Fractal sets may be constructed in many different ways. Barnsley [3] singled out the “iterated function system” method: The fractal set K is made up of parts, each of which is a shrunken copy of the whole set. Mauldin and Williams [13] provided a more general setting, where several sets Kv are involved, each of them is made up of parts, and each part is a shrunken copy of one of the parts (the same one or a different one). The combinatorics of the way in which the parts fit together is described by a directed multigraph. This is described in detail below (Definition 1.5). In addition to Mauldin and Williams, compare “recurrent iterated function system” [3, Ch. X], “Markov self-similar sets” [20], and “mixed self-similar systems” [1]; see also [5], [19]. In the text [6, Section 6.4] there is an exposition of the Mauldin–Williams computation of the dimension of the attractors for a graph-directed iterated function system consisting of similarities. “Similarities” are functions f : S → T between metric spaces that satisfy equations of the form (0.1) d ( f(x),f(y) ) = rd(x,y) for all x, y ∈ S.