In this note we introduce a notion of morphism between two hyperbolic iterated function systems. We prove that the graph is attractor an system, giving Closed Graph Theorem, and show how it can be used to approach topological conjugacy problem for

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...