Abstract
A change to the Szegö matrix recurrence relation, satisfied by orthonormal poly- nomials on the unit circle, gives rise to a linear map by the action of matrices belonging to the group SU (1; 1). The companion factorization of such matrices, via 2nd-order linear homogeneous difference equations, provides a compact representation of the orthogonal polynomial on the circle. Moreover, an isomorphism SU (1; 1) ≃ SL(2; R) enables the introduction of a linear non-autonomous area-preserving map. This dynamical system has counterparts in those from the complex Szegö recurrence relation, and some basic results are outlined.
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