An Extension of Recurrent Iterated Function System— From the Viewpoint of Graph Theory and Product Space∗

Fractal Interpolation Function is the attractor of Iterated Function System (IFS), to overcome the weakness of IFS that IFS only reflect the similarity between global region and local region of a graph, Barnsley put forward the Recurrent Iterated Function System and the Recurrent Fractal Interpolation Function. RIFS reflect the similarity between local and local, it can make more complicated graph. In his paper, Barnsley strictly proved the existence and uniqueness of the attractor of RIFS, but he didn’t give the relationship between RIFS and Recurrent Fractal Interpolation Function. Based on his research, we extended the concept of RIFS and prove the existence and uniqueness of it’s attractor. Further more, we can see that under our extended definition, the Recurrent Fractal Interpolation Function can be seen as the attractor of a RIFS.