The induced contractive maps on the covering spaces

Let ( ˜x , p) be the universal covering space of a compact metrizable space x, which is compact and locally path connected. in this paper, we show that there exist metrics d and d ′for x and ˜x, respectively, such that any contractive map f : x → x induces a contractive map on ˜x. as an application, it is obtained that every iterated function system(ifs) on the space x with attractor k, induces an ifs on ˜x with attractor ˜k , such that p( ˜k ) = k.