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We consider multi-valued maps defined on a complete gauge space endowed with a directed graph. We establish a fixed point result for maps which send connected points into connected points and satisfy a generalized contraction condition. Then, we study infinite graph-directed iterated function systems (H-IIFS). We give conditions insuring the existence of a unique attractor to an H-IIFS. Finally, we apply our fixed point result for multi-valued contractions on gauge spaces endowed with a graph to obtain more information on the attractor of an H-IIFS. More precisely, we construct a suitable gauge space endowed with a graph G and a suitable multi-valued G-contraction such that its fixed points are sub-attractors of the H-IIFS.

A contraction principle on gauge spaces with graphs and application to infinite graph-directed iterated function systems