Transversal family of non-autonomous conformal iterated function systems

Abstract

We study Non-autonomous Iterated Function Systems (NIFSs) with overlaps. A NIFS on a compact subset X\subset\mathbb{R}^{m} is a sequence \Phi=(\{\phi^{(j)}_{i}\}_{i\in I^{(j)}})_{j=1}^{\infty} of collections of uniformly contracting maps \phi^{(j)}_{i}\colon X\rightarrow X , where I^{(j)} is a finite set. In comparison to usual iterated function systems, we allow the contractions \phi^{(j)}_{i} applied at each step j to depend on j . In this paper, we focus on a family of parameterized NIFSs on \mathbb{R}^{m} . Here, we do not assume the open set condition. We show that if a d -parameter family of such systems satisfies the transversality condition, then for almost every parameter value the Hausdorff dimension of the limit set is the minimum of m and the Bowen dimension. Moreover, we give an example of a family \{\Phi_{t}\}_{t\in U} of parameterized NIFSs such that \{\Phi_{t}\}_{t\in U} satisfies the transversality condition but \Phi_{t} does not satisfy the open set condition for any t\in U .