Separation Properties For Self-affine Graph-directed Sets

For IFS consisting of linear contractions, A.Shief illustrates the relations amongopen set condition, strong open set condition and positive and finite Hausdor? measure.The result is important and fundamental in fractal geometry. It has been shown theyare still true for conformal IFS.He and Lau generalized the result to a?ne IFS with a single matrix. To overcomethe di?culties coming from the di?erent contraction ratios in di?erent directions, theyemployee a pseudo-norm determined by the matrix.In this paper, we generalize the result to graph IFS with a single matrix. Thesystem consists of a directed graph G = (V,Γ), and mappings of the form:φe(x) =A?1(x + de), where A is a d×d expanding matrix and de∈Rd.We prove that for graph IFS of a single matrix, open set condition, strong openset condition and positive finite Hausdor? measure are all equivalent. Our results findapplications in the study of atomic surfaces of hyperbolic substitutions.