  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 $\mathbf{P}$.
