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

the reciprocal degree distance (rdd)‎, ‎defined for a connected graph $g$ as vertex-degree-weighted sum of the reciprocal distances‎, ‎that is‎, ‎$rdd(g) =sumlimits_{u,vin v(g)}frac{d_g(u)‎ + ‎d_g(v)}{d_g(u,v)}.$ the reciprocal degree distance is a weight version of the harary index‎, ‎just as the degree distance is a weight version of the wiener index‎. ‎in this paper‎, ‎we present exact formu...

For given closed orientable 3-manifolds M and N let D(M,N) be the set of mapping degrees from M to N . We address the problem: For which N , D(M,N) is finite for all M? The answer is known in Thurston’s picture of closed orientable irreducible 3-manifolds unless the target is a non-trivial graph manifold. We prove that for each closed non-trivial graph manifold N , D(M,N) is finite for all grap...