Second angular derivatives and parabolic iteration in the unit disk

Contreras, Manuel; Díaz-Madrigal, Santiago; Pommerenke, Christian

doi:10.1090/S0002-9947-09-04873-9

Second angular derivatives and parabolic iteration in the unit disk
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by Manuel D. Contreras, Santiago Díaz-Madrigal and Christian Pommerenke PDF

Trans. Amer. Math. Soc. 362 (2010), 357-388 Request permission

Abstract:

In this paper we deal with second angular derivatives at Denjoy-Wolff points for parabolic functions in the unit disc. Namely, we study and analyze the existence and the dynamical meaning of this second angular derivative. For instance, we provide several characterizations of that existence in terms of the so-called Koenigs function. It is worth pointing out that there are two quite different classes of parabolic iteration: those with positive hyperbolic step and those with zero hyperbolic step. In the first case, the Koenigs function is in the Carathéodory class but, in the second case, it is even unknown if it is normal. Therefore, the ideas and techniques to approach these two cases are really different. In the end, we also present several rigidity results related to the second angular derivatives at Denjoy-Wolff points.

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