Finite automata as dynamical systems

Question

In abstract (deterministic finite) automata theory the set of states of an automaton is an arbitrary set Q, and the transistion function is a specific set δ ⊆ Q × Σ × Q (with alphabet Σ, i.e. another set).

In a "real" automaton - seen as a dynamical system - its states are configurations of the system. These configurations may determine the transition function - seen as a dynamical process - in a traceable way.

I am looking for references where the states of an deterministic automaton are treated or at least discussed as configurations of a dynamical system - and not only as arbitrary atomic entities (symbols). Where should I look at, how could I search?

Answer 1

You might be interested in the following papers. I suggest you to read [2] first: it is shorter and more elementary than [1].

[1] Béal, Marie-Pierre; Perrin, Dominique. Symbolic dynamics and finite automata. Handbook of formal languages, Vol. 2, 463--505, Springer, Berlin, 1997.

[2] Perrin, Dominique. Symbolic dynamics and finite automata. Mathematical foundations of computer science 1995 (Prague), 94--104, Lecture Notes in Comput. Sci., 969, Springer, Berlin, 1995.

Answer 2

This may not be quite what you're looking for, but graph dynamical systems apply dynamical systems theory to automata theory. Particularly, cellular automata. There is a book on this: http://www.amazon.com/Introduction-Sequential-Dynamical-Systems-Universitext/dp/0387306544

Much of the research in this field comes from the Network Dynamics and Simulation Science Lab (NDSSL) at Virginia Tech. Henning Mortveit and Chris Barrett are good names for literature.