The Linear Fractional Model Theorem and Aleksandrov–Clark measures

A remarkable result by Denjoy and Wolff states that every analytic self‐map φ of the open unit disc D of the complex plane, except an elliptic automorphism, has an attractive fixed point to which the sequence of iterates{ φ n } n ⩾ 1converges uniformly on compact sets: if there is no fixed point in D , then there is a unique boundary fixed point that does the job, called the Denjoy–Wolff point . This point provides a classification of the analytic self‐maps of D into four types: maps with interior fixed point, hyperbolic maps, parabolic automorphism maps and parabolic non‐automorphism maps. We determine the convergence of the Aleksandrov–Clark measures associated to maps falling in each group of such classification.