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For any contractive iterated function system (IFS, including the Moransystems), we show that there is a natural hyperbolic graph on the symbolicspace, which yields the H\"{o}lder equivalence of the hyperbolic boundary andthe invariant set of the IFS. This completes the previous studies (\cite{[Ka]}, \cite{[LW1]}, \cite{[W]}) by eliminating superfluous conditions, andadmits more classes of sets (e.g., the Moran sets). We also show that thebounded degree property of the graph can be used to characterize certainseparation properties of the IFS (open set condition, weak separationcondition); the bounded degree property is particularly important when weconsider random walks on such graphs. This application and the otherapplication to Lipschitz equivalence of self-similar sets will be discussed.

On Hyperbolic graphs induced by iterated function systems