On selfsimilar Jordan curves on the plane

We study the attractors of a finite system of planar contraction similarities $S_j$ $(j=1,\dots,n)$ satisfying the coupling condition: for a set $\{x_0,\dots,x_n\}$ of points and a binary vector $(s_1,\dots,s_n)$, 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.