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 X^m is a sequence = (\^{ (j) ₈\}₈ ₈^ (₉) ) ₉=₁^ of collections of uniformly contracting maps ^ (j) ₈ X X, where I^ (j) is a finite set. In comparison to usual iterated function systems, we allow the contractions ^ (j) ₈ applied at each step j to depend on j. In this paper, we focus on a family of parameterized NIFSs on 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 \ₓ\ₓ ₔ of parameterized NIFSs such that \ₓ\ₓ ₔ satisfies the transversality condition but ₓ does not satisfy the open set condition for any t U.