A map u from a linearly ordered finite set into itself induces a matrix A(a). If u is a cyclic permutation, A(o) is either primitive or reducible. In the latter case the characteristic polynomial of A(u) has a product decomposition. If u is monotonic, then both factors arise from the characteristic polynomials of induced permutations.
There has been a great deal of work done in recent years on the properties of continuous endomorphisms of closed intervals when these are viewed as dynamical systems [8, 10, 131. This paper looks at some questions in linear algebra which are related to and motivated by the theory of one-dimensional dynamical systems. In this theory important objects of study are the finite invariant sets of the generating function f, in particular the orbits of periodic points [i.e. points x with f"(x) = x]. Such a set S partitions the domain of f into subintervals. Using the natural order on S, the behavior of f on S generates a directed graph with associated O-l matrix A (for details see [13]). These can be used to determine many properties of the dynamical system. In particular, the spectral radius of A is related to the topological entropy, a key invariant of the dynamical system [6, 7, 111.
The essential ingredients here are a finite ordered set and an endomorphism of this set. From the viewpoint of dynamical systems the most