Profiles of dynamical systems and their algebra

The commutative semiring $\mathbf{D}$ of finite, discrete-time dynamical systems was introduced in order to study their (de)composition from an algebraic point of view. However, many decision problems related to solving polynomial equations over $\mathbf{D}$ are intractable (or conjectured to be so), and sometimes even undecidable. In order to take a more abstract look at those problems, we introduce the notion of "topographic" profile of a dynamical system $(A,f)$ with state transition function $f \colon A \to A$ as the sequence $\mathop{\mathrm{prof}} A = (|A|_i)_{i \in \mathbb{N}}$, where $|A|_i$ is the number of states having distance $i$, in terms of number of applications of $f$, from a limit cycle of $(A,f)$. We prove that the set of profiles is also a commutative semiring $(\mathbf{P},+,\times)$ with respect to operations compatible with those of $\mathbf{D}$ (namely, disjoint union and tensor product), and investigate its algebraic properties, such as its irreducible elements and factorisations, as well as the computability and complexity of solving polynomial equations over $\mathbf{P}$.