Essential angular derivatives and maximum growth of Koenigs eigenfunctions

We introduce the notion of essential angular derivative for functions phi m apping the open unit disk U holomorphically into itself. After exploring so me of its basic properties, we show how the essential angular derivative of phi determines the maximum growth rate of the Koenigs eigenfunction sigma for phi when phi has an attractive fixed point in U Our work answers some q uestions about growth of Koenigs functions recently posed by Pietro Poggi-C orradini. (C) 1998 Academic Press.