Title: Iterated graph neural network system. Abstract: We present Iterated Graph Neural Network System (IGNNS), a new framework of Graph Neural Networks (GNNs), which can deal with undirected graph and directed graph in a unified way. The core component of IGNNS is the Iterated Function System (IFS), which is an important research field in fractal geometry. The key idea of IGNNS is to use a pair of affine transformations to characterize the process of message passing between graph nodes and assign an adjoint probability vector to them to form an IFS layer with probability. After embedding in the latent space, the node features are sent to IFS layer for iterating, and then obtain the high-level representation of graph nodes. We also analyze the geometric properties of IGNNS from the perspective of dynamical system. We prove that if the IFS induced by IGNNS is contractive, then the fractal representation of graph nodes converges to the fractal set of IFS in Hausdorff distance and the ergodic representation of that converges to a constant matrix in Frobenius norm. We have carried out a series of semi supervised node classification experiments on citation network datasets such as citeser, Cora and PubMed. The experimental results show that the performance of our method is obviously better than the related methods.