ABSTRACT: An increasing body of experimental evidence supports the belief that random behavior observed in a wide variety of physical systems is due to underlying deterministic dynamics on a low dimensional chaotic attractor. The behavior exhibited by a chaotic attractor is predictable on short time scales and unpredictable (random) on long time scales. The unpredictability, and so the attractor's degree of chaos, is effectively measured by the entropy. Symbolic dynamics is the application of information theory to dynamical systems. It provides experimentally applicable techniques to compute the entropy, and also makes precise the difficulty of constructing predictive models of chaotic behavior. Furthermore, symbolic dynamics offers methods to distinguish the features of different kinds of randomness that may be simultaneously present in any data set: chaotic dynamics, noise in the measurement process, and fluctuations in the environment.
In this paper, we first review the development of symbolic dynamics given in Symbolic Dynamics of One-Dimensional Maps: Entropies, Finite Precision, and Noise. In the later sections, we report new results on a scaling theory of symbolic dynamics in the presence of fluctuations. We conclude with a brief discussion of the experimental application of these ideas.