Abstract : We introduce an algebraic approach for the analysis and composition of finite, discrete-time dynamical systems in terms of the category-theoretical operations of sum (disjoint union) and (tensor) product, which correspond to alternative and synchronous execution. This defines a semiring structure over the set of dynamical systems (modulo isomorphism), which allows us to express decomposition problems in terms of factorisation and polynomial equations. We analyse the algebraic properties of the semiring and prove that most dynamical systems are irreducible, and that the reducible ones sometimes admit multiple factorisation into irreducibles. We also prove that polynomial equations over this semiring are, in general, algorithmically unsolvable, and that many solvable subclasses of equations, including linear ones, are intractable even if the explicit dynamics of the systems is given as input (while in applications, where the dynamical systems are described more succinctly, these problems are conjecturally even harder). Finally, we also analyse dynamical systems in terms of their “topographic profiles”, a more abstract view in terms of the number of states at a given distance from the limit cycles of the system, which turns out to share many of the same algebraic properties.