Stable sheets in four dimensions
Four zipper families in exterior parameters |c₁|, |c₂| > 1
Stability here means embeddedness of the canonical marked zipper within each two-real-dimensional family. These sheets lie in the four-dimensional parameter space; they do not classify an open region of that space. Parameters outside a stable zipper set can still have connected attractors.
4,096 certified cells around the island probe, magnified in the c₁ plane. The circular C marker in the atlas locates this patch; its symbol size is independent of the patch size.
Axes show c₁ − c₁(C), in units of 10⁻⁶.
Each sheet approximates the nonlinear image of a certified inner cover with a triangular mesh. Color and the weight filter retain the fourth coordinate w; the three-dimensional view projects it out. Blank space carries no classification. Apparent intersections between projected sheets need not be intersections in four dimensions.
Coordinates of the projection
θ₁ = arg(a), θ₂ = arg(b).
X = (4 + ρ cos θ₁) cos θ₂
Y = (4 + ρ cos θ₁) sin θ₂
Z = ρ sin θ₁
At fixed w and orientation, this chart is injective on the displayed range 0 < ρ ≤ 2. DD and DO belong to separate orientation charts. Its central hole comes from the display map. The faint grid is the reference torus ρ = 2. The slider selects a finite band w ± 0.025 around the centres of displayed cells.
Three probes, four families
Select a curve to locate itCurve branches are red and blue where a stable certificate is available. The two neutral C examples are included for comparison; they are not part of the displayed stable sheets. These polygonal drawings have a uniform error bound below 0.0051 in the endpoint coordinate.