Abstract:
We study dissipative rotational attractors in two settings: Siegel
disks of Hénon maps and minimal attractors of diffeomorphisms of the
annulus. Jointly with D. Gaydashev, we extend renormalization of
Siegel maps and critical circle maps to small 2D perturbations, and
use renormalization tools to study the geometry of the attractors. In
the Siegel case, jointly with D. Gaydashev and R. Radu we prove that
for sufficiently dissipative Hénon maps with semi-Siegel points with
golden-mean rotation angles, Siegel disks are bounded by
(quasi)circles. In the annulus case, jointly with D. Gaydashev, we
prove that for bounded type rotation number, "critical" annulus maps
have a minimal attractor which is a C0, but not smooth, circle --
answering a question of E. Pujals.