Transversal family of non-autonomous conformal iterated function systems

We study Non-autonomous Iterated Function Systems (NIFSs) with overlaps. A NIFS on a compact subset Xᵐ is a sequence Φ=(\{φ⁽ʲ⁾ᵢ\}_{i∈ I⁽ʲ⁾})ⱼ₌₁∞ of collections of uniformly contracting maps φ⁽ʲ⁾ᵢ X→ X , where I⁽ʲ⁾ is a finite set. In comparison to usual iterated function systems, we allow the contractions φ⁽ʲ⁾ᵢ applied at each step j to depend on j . In this paper, we focus on a family of parameterized NIFSs on Rᵐ . 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 \{Φₜ\}t∈ U of parameterized NIFSs such that \{Φₜ\}t∈ U satisfies the transversality condition but Φₜ does not satisfy the open set condition for any t∈ U .