Systems and control theory have found wide application in the analysis and design of numerical algorithms. The authors present an equivalent discrete-time dynamical system interpretation of an algorithm commonly used in information theory called belief propagation (BP). BP is one instance of the so-called sum–product algorithm and arises, for example, in the context of iterative decoding of low-density parity-check codes. The authors review a few known results from information theory in the language of dynamical systems and show that the typically very high-dimensional, non-linear dynamical system corresponding to BP has interesting structural properties. For the linear case, they completely characterise the behaviour of this dynamical system in terms of its asymptotic input–output map. Finally, the authors state some of the open problems concerning BP in terms of the dynamical system presented.