#### Title

A Generalized Finite Type Condition for Iterated Function Systems

#### Document Type

Article

#### Publication Date

1-30-2007

#### Publication Title

Advances in Mathematics

#### DOI

10.1016/j.aim.2006.03.007

#### ISSN

0001-8708

#### Abstract

We study *iterated function systems* (IFSs) of contractive similitudes on *R*^{d} with overlaps. We introduce a generalized finite type condition which extends a more restrictive condition in [S.-M. Ngai, Y. Wang, Hausdorff dimension of self-similar sets with overlaps, J. London Math. Soc. (2) 63 (3) (2001) 655–672] and allows us to include some IFSs of contractive similitudes whose contraction ratios are not exponentially commensurable. We show that the generalized finite type condition implies the weak separation property. Under this condition, we can identify the attractor of the IFS with that of a graph-directed IFS, and by modifying a setup of Mauldin and Williams [R.D. Mauldin, S.C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988) 811–829], we can compute the Hausdorff dimension of the attractor in terms of the spectral radius of certain weighted incidence matrix.

#### Recommended Citation

Lau, Ka-Sing, Sze-Man Ngai.
2007.
"A Generalized Finite Type Condition for Iterated Function Systems."
*Advances in Mathematics*, 208 (2): 647-671.
doi: 10.1016/j.aim.2006.03.007 source: https://www.sciencedirect.com/science/article/pii/S0001870806000958?via%3Dihub

https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/610