Iterated function systems with super-exponentially close cylinders II

Until recently, it was an important open problem in Fractal Geometry to determine whether there exists iterated function system acting on $\mathbb {R}$ with no exact overlaps for which cylinders are super-exponentially close at all small scales. Iterated systems satisfying these properties were shown exist by the author and Bárány Käenmäki. In this paper we prove a general theorem existence of such within parameterised family. This shows that if family contains two independent subfamilies, set parameters cause satisfies some weak topological assumptions, then original will contain desired properties. We include several explicit examples families can be applied.