Traces of algebraic integers and dynamical systems

We study the number of periodic points in symbolic dynamical systems; we prove the following formula, which concerns square matrices with entries in Z:∑d|nμ(d)TrAn/d≡0modn.Here μ(d) is the Möbius function. We ask some questions about the sequence (TrAn)n⩾0. We also give a language-theoretic proof of a result of Boyle and Handelman, concerning the spectrum of positive matrices.