2. In a long history associated with the problem on iterating holomorphic mappings and their fixed points, the work of G. Julia, J. Wolff and C. Caratheodory are among the most important. Using their results, one can describe the behavior of iterates of a holomorphic self-mapping of the unit disk near its fixed point. These iterations form a discrete-time semigroup on the disk. We study the asymptotic behavior of a continuous-time semigroup on the unit ball of a complex Hilbert space. We assume that a semigroup under consideration consists of nonexpansive mappings with respect to the hyperbolic metric which are not necessarily holomorphic.

3. The Julia-Wolff-Caratheodory Theorem z τ z τ F(z) F

5. Dynamic extension of the Julia-Wolff-Caratheodory Theorem For attractive fixed points – M. Elin and D. Shoikhet (2001) For arbitrary fixed points – M. D. Contreras, S. Díaz-Madrigal and Ch. Pommerenke (2005)

11. z z

12. Thank you for your attention!!!