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

G. A. Edgar & Jeffrey Golds

ABSTRACT. Suppose a graph-directed iterated function system consists of maps f_e with upper estimates of the form $d\big(f_e(x),f_e(y)\big) \le r_e d(x,y)$ . Then the fractal dimension of the attractor K_v of the IFS is bounded above by the dimension associated to the Mauldin-Williams graph with ratios r_e. Suppose the maps f_e also have lower estimates of the form $d\big(f_e(x),f_e(y)\big) \ge r'_e \,d(x,y)$ and that the IFS also satisfies the strong open set condition. Then the fractal dimension of the attractor K_v of the IFS is bounded below by the dimension associated to the Mauldin-Williams graph with ratios r'_e. When r_e = r'_e, then the maps are similarities and this reduces to the dimension computation of Mauldin and Williams for that case.

Department of Mathematics
The Ohio State University
Columbus, Ohio 43210

Also: Silicon Graphics
1600 Amphitheatre Parkway
Mountain View, California 94043
EMAIL: edgar@math.ohio-state.edu (G. A. Edgar)
EMAIL: golds@csd.sgi.com (Jeffrey Golds)

Submitted: June 29, 1999; revised: December 4, 1998.