# Meeting Details

Title: Dynamics of dissipative polynomial automorphisms of C^2 Center for Dynamics and Geometry Colloquium Mikhail Lyubich, SUNY Stony Brook Two-dimensional complex dynamics displays a number of phenomena that are not observable in dimension one. However, if f is moderately dissipative then there are deeper similarities between the two fields. In particualar, a nearly complete classification of periodic Fatou components has been recently obtained: Theorem 1 (joint with Han Peters): Any periodic component of the Fatou set is either an attracting basin or parabolic basin, or the basin of a rotation domain (Siegel disk or Herman ring). In complex and real one-dimensional world, structurally stable maps are dense. In dimension two this fails because of the Newhouse phenomenon caused by homoclinic tangencies. Palis conjectured that in the real two-dimensional case this is the only reason for failure. We prove a complex version of this conjecture: Theorem 2 (joint with Romain Dujardin): Any moderately dissipative polynomial automorphism of C^2 is either weakly stable" or it can be approximated by a map with homoclinic tangency.