Beside its theoretical interest, this was motivated by the increasing importance of infinite state systems in computer science.Īny attempt at studying symbolic dynamics on infinite alphabets has to deal with the fact that the discrete topology, employed in the finite case, leads to shift spaces which are not compact. This note is an attempt to generalize this meeting ground of topology, graph theory, and spectral theory to infinite alphabets, especially countably infinite alphabets. On an exponential scale, this rate equals the growth of the number of finite words.
Considering the graph as a linear map, the spectral radius measures the rate of dilation under iterated application. In the case of a topological Markov chain, the entropy equals the natural logarithm of the spectral radius of the generating graph. For symbolic systems, the entropy equals the exponential growth rate of the number of finite words of fixed length. The most important numerical invariant of dynamical systems is the topological entropy. In computer science, certain symbolic systems, namely, the topological Markov chains generated by finite graphs, model the evolution of finite transition systems, and the class of sofic symbolic systems (factors of topological Markov chains) models the evolution of certain automata. Symbolic dynamical systems on finite alphabets are classical mathematical objects that provide a wealth of examples and have greatly influenced theoretical developments in dynamical systems.
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