The Szegö matrix recurrence and its associated linear non-autonomous area-preserving map

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 2^nd-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.