SEPARATION PROPERTIES FOR BI-LIPSCHITZ ITERATED FUNCTION SYSTEMS

Abstract

In this paper, we study the iterated function systems (IFSs) consisting of bi-Lipschitz mappings instead of conformal contractions, focusing on IFSs that do not satisfy the open set condition. We define a weak* separation condition (W*SC), which is strictly weaker than the weak separation condition of Lau and Ngai. By assuming the bounded distortion property, we show that the W*SC is equivalent to the identity limit criterion of Zerner for such IFSs. In particular, in the one-dimensional case, we show that the W*SC is also equivalent to Ahlfors–David regularity, the Assouad dimension is strictly less than [Formula: see text] and positivity of the [Formula: see text]-dimensional Hausdorff measure, where [Formula: see text] is the zero of the topological pressure.