A fractal dimension estimate for a graph-directed IFS of non-similarities
Abstract
Suppose a graph-directed iterated function system consists of maps fe with upper estimates of the form d(fe(x),fe(y)) <= re d(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'e d(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 & Williams for that case.
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.