Information flow and Lyapunov exponent on coupled map lattice The information flow is studied in many fields, not only network theory but also meteorology, economics and so on. In such fields, the system on which information is transported is often modelled as a dynamical system which consists of the source part, the receiver part and the other parts. We regard the state of the source part at a certain time as the input, and the information of the input is transported to the receiver part. However, the dynamical system cannot transport information perfectly. Uncertainty is one of the reasons of the imperfect information flow. Unobserved states, which are states of the other parts for instance, are uncertainty on the system. External noise is also included in the uncertainty if it is added to the system. The dynamical system is characterized by the Lyapunov exponent, and hence it is interesting to investigate the relation between the information flow and the Lyapunov exponent. If the Lyapunov exponent of the system is too small, difference of the inputs damps in the way of transportation and hence no correlation appears between the inputs and the receiver states. If the Lyapunov exponent of the system is too large, the receiver state responds sharply to the uncertainty. We, therefore, conjecture that the system does not transport information when the Lyapunov exponent is too small or too large and amount of information flow takes a peak value when the Lyapunov exponent is one middle value. In addition, we are also interested in the change of the information flow in the view point from the Lyapunov exponent when we add the noise to the system. We calculate the information flow and the Lyapunov exponent on a 1-dimentional coupled map lattice numerically, and verify the conjecture. The sign of the Lyapunov exponent is negative when the amount of the information flow is maximized. The negative sign is observed in the two types of the coupled map lattices with the logistic map and with the tent map. We also show that, depending on the system parameter values, the noise added to a lattice point increases or decreases the information flow when the Lyapunov exponent is around the middle value. If both the source part and the receiver part are coupled with the noisy lattice point, it is no wonder that the amount of the information flow increses because both the parts share the common noise. One of remarkable findings is that the amount of the information may increase even when the noisy lattice point couples with the receiver part only.