Irreducible Complexity of Iterated Symmetric Bimodal Maps

The concept of irreducible complexity of a biological system was introduced by Behe [1] in 1996. His point of view is that an organism consisting of a finite, possibly very large, number of independent components, coupled together in some way, exhibits irreducible complexity if, by removing any of its components, the reduced system no longer functions meaningfully. Using the language of nonlinear dynamics and chaos theory, Boyarsky and Góra [2] reinterpreted Behe’s definition from a Markov transition matrix perspective by saying that a system is irreducibly complex if the associated transition matrix is primitive but no principal submatrix is primitive. It is our conviction that the concept of reducible complexity of a dynamical system can also be interpreted in terms of a factorization: within Milnor and Thurston’s kneading theory framework and the topological classification obtained from it, Derrida et al. [4] introduced a ∗-product between unimodal kneading sequences for which it was possible to prove that the topological entropy, a measure of complexity, of a factorizable system is equal to the topological entropy of one of the factors. Despite of a larger number of its components, the complexity of the system remains the same whenever its irreducible component, a factor of the product, does not change. Some years later, Lampreia et al. [5] introduced a Markov transition matrix formalism associated with the kneading theory and a product between unimodal matrices corresponding to the Derrida-Gervois-Pomeau ∗-product. Then they proved that irreducible unimodal kneading sequences correspond to primitive Markov transition matrices.