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