Symbolic dynamics is a rapidly growing part of dynamical systems. Although it originated as a method to study general dynamical systems, the techniques and ideas have found significant applications in data storage and transmission as well as linear algebra. This is the first general textbook on symbolic dynamics and its applications to coding, and we hope that it will stimulate both engineers and mathematicians to learn and appreciate the subject.

Dynamical systems originally arose in the study of systems of differential equations used to model physical phenomena. The motions of the planets, or of mechanical systems, or of molecules in a gas can be modeled by such systems. One simplification in this study is to discretize time, so that the state of the system is observed only at discrete ticks of a clock, like a motion picture. This leads to the study of the iterates of a single transformation. One is interested in both quantitative behavior, such as the average time spent in a certain region, and also qualitative behavior, such as whether a state eventually becomes periodic or tends to infinity. Symbolic dynamics arose as an attempt to study such systems by means of discretizing space as well as time. The basic idea is to divide up the set of possible states into a finite number of pieces, and keep track of which piece the state of the system lies in at every tick of the clock.