Abstract
Consider an operator X satisfying the algebraic relation XX ∗ = f ( X ∗ X ), where f is the one-parameter family of quadratic maps f b ( x ) = 4 bx (1 − x ) with b ∈ [0, 1]. There is a correspondence between the periodic orbits of the dynamical system ([0, 1], f b ) 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.
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