Introduction to discrete holomorphic dynamic

In this course we will give an introduction to some aspects of holomorphic dynamical systems with emphasis on the « discrete side ». The focus of the course will be on the dichotomy between local and global dynamical phenomena related to the iteration of holomorphic endomorphisms in complex dimension 1, and on the relations between the local and the global aspect. If time allows, we will also deal with the higher dimensional case. Several research problems will be discussed during the course. Outline:

• Iteration in the complex plane and the Riemann sphere. – Fatou and Julia sets. – The polynomial case. – Invariance. – Conjugaison. – Periodic and critical points. – Description of Julia sets. – Description Fatou sets.
• Local dynamics. – Linear case. – Conjugaisons locales. – Fixed points geometricaly attracting and repulsing. – Böttcher’s Theorem and polynomial dynamics. – Parabolic fixed points: Théorème de la Fleur de Leau-Fatou. – Elliptic fixed points: stability and linearability, Cremer points and Siegel discs. Higher dimension.
• References: Dynamics in one complex variable. Third edition. John Milnor, Annals of Mathematics Studies, 160. Princeton University Press, Princeton, NJ, 2006. viii+304 pp. Dynamics in One Complex Variable by Xavier Buff and John H. Hubbard

References: Dynamics in one complex variable. Third edition. John Milnor, Annals of Mathematics Studies, 160. Princeton University Press, Princeton, NJ, 2006. viii+304 pp. Dynamics in One Complex Variable by Xavier Buff and John H. Hubbard