Title: Transforming Recurrent Neural Networks with Attention and Fixed-point Equations. Abstract: Transformer has achieved state of the art performance in multiple Natural Language Processing tasks recently. Yet the Feed Forward Network(FFN) in a Transformer block is computationally expensive. In this paper, we present a framework to transform Recurrent Neural Networks(RNNs) and their variants into self-attention-style models, with an approximation of Banach Fixed-point Theorem. Within this framework, we propose a new model, StarSaber, by solving a set of equations obtained from RNN with Fixed-point Theorem and further approximate it with a Multi-layer Perceptron. It provides a view of stacking layers. StarSaber achieves better performance than both the vanilla Transformer and an improved version called ReZero on three datasets and is more computationally efficient, due to the reduction of Transformer's FFN layer. It has two major parts. One is a way to encode position information with two different matrices. For every position in a sequence, we have a matrix operating on positions before it and another matrix operating on positions after it. The other is the introduction of direct paths from the input layer to the rest of layers. Ablation studies show the effectiveness of these two parts. We additionally show that other RNN variants such as RNNs with gates can also be transformed in the same way, outperforming the two kinds of Transformers as well.