  This paper introduces the \textit{truncator} map as a dynamical system on the
space of configurations of an interacting particle system. We represent the
symbolic dynamics generated by this system as a non-commutative algebra and
classify its periodic orbits using properties of endomorphisms of the resulting
algebraic structure. A stochastic model is constructed on these endomorphisms,
which leads to the classification of the distribution of periodic orbits for
random truncator maps. This framework is applied to investigate the periodic
transitions of Bornholdt's spin market model.
