Integrality of matrices, finiteness of matrix semigroups, and dynamics of linear cellular automata
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Let K be a finite commutative ring, and let L be a commutative K-algebra. Let A and B be two n × n-matrices over L that have the same characteristic polynomial. The main result of this paper states that the set { A^0,A^1,A^2,...} is finite if and only if the set { B^0,B^1,B^2,...} is finite. We apply this result to the theory of discrete time dynamical systems. Indeed, it gives a complete and easy-to-check characterization of sensitivity to initial conditions and equicontinuity for linear cellular automata over the alphabet K^n for K = Z/mZ, i.e. cellular automata in which the local rule is defined by n× n-matrices with elements in Z/mZ. To prove our main result, we derive an integrality criterion for matrices that is likely of independent interest. Namely, let K be any commutative ring (not necessarily finite), and let L be a commutative K-algebra. Consider any n × n-matrix A over L. Then, A ∈L^n × n is integral over K (that is, there exists a monic polynomial f ∈K[t] satisfying f(A) = 0) if and only if all coefficients of the characteristic polynomial of A are integral over K. The proof of this fact relies on a strategic use of exterior powers (a trick pioneered by Gert Almkvist).
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