On selfsimilar Jordan curves on the plane

We study the attractors of a finite system of planar contraction similarities S-j (j = 1,. . ., n) satisfying the coupling condition: for a set {x(0),...,x(n)} of points and a binary vector (s(1),...,s(n)), called the signature, the mapping S-j takes the pair {x(0), x(n)} either into the pair {x(j-1), x(j)} (if s(j) = 0) or into the pair {x(j), x(j-1)} (if s(j) = 1). We describe the situations in which the Jordan property of such attractor implies that the attractor has bounded turning, i.e., is a quasiconformal image of an interval of the real axis.