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. Gaidashev 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.