Iterations, Denjoy-Wolff Points and Ergodic Properties of Point-Mass Singular Inner Functions

We consider the singular inner function Mζ, α whose singular measure is the point mass α > 0 at ζ on the unit circle. We give the explicit integral representaion of the (n + 1)th iterate Mn +1ζ, α and determine the location of the Denjoy-Wollf point of Mζ, α. As applications we also discuss some ergodic properties of the restriction of Mζ, α on the unit circle.