Considering travelers adjust their travel path from day to day, we propose a discrete dynamical system model.It is built upon route-based variables to describe the evolutionary process that traffic flow achieves equilibrium from a non-equilibrium state.And its stationary state is equivalent to the Logit stochastic user equilibrium state.We also analyze several properties of the model, including the equivalence between the stationary state and user equilibrium state and the uniqueness of its stationary state.The model is given in general form and hence we introduce aspecial case of the model.Finally, numerical experiments are carried out to show application and properties of the model.