The Linear Fractional Model Theorem and Aleksandrov-Clark measures

Abstract

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 ⩾ 1 converges 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.