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 $\Phi =(\{{\varphi }_i^{(j)}{\}}_{i\in I^{(j)}}{)}_{j=1}^{\infty }$ of collections of uniformly contracting maps ${\varphi }_i^{(j)}:X\to X$, where $I^{(j)}$ is a finite set. In comparison to usual iterated function systems, we allow the contractions ${\varphi }_i^{(j)}$ 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 $t\in U$.