Profiles of dynamical systems and their algebra
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The commutative semiring π 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 π 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 βtopographicβ profile of a dynamical system (A,f) with state transition function f A β A as the sequence prof A = (|A|_i)_i ββ, 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 (π,+,Γ) with respect to operations compatible with those of π (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 π.
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