Finite dimensional representations of *-algebras arising from a quadratic map

Consider an operator X satisfying the algebraic relation XX* = f(X*X), where f is the one-parameter family of quadratic maps fb(x) = 4bx(1 − x) with b [0, 1]. There is a correspondence between the periodic orbits of the dynamical system ([0, 1], fb) and the unitary classes of matrices X satisfying XX* = f(X*X). Using the symbolic dynamics theory for interval maps, we describe in detail the combinatorial structure of the finite dimensional representations associated to this relation.