Parameter families
Placement in the 4D atlas
Projection & clipping
Colour identifies families. Surface geometry projects stored stable covers; thin lines show analytic sections. Projection may identify different parameters.
Tip-point sets
Families, parameters and coded points
Move a locator to change the attractor. Select a coded point to explore its tip-point set.
Loading catalogue…
Selected family
Selected family
Exterior parameter slice
Locator A
Attractor and coded points
F₀(K)F₁(K)
Equation, exact record & proof
Definitions
A stable set is relative to its prescribed contacts
For a parameter family R with prescribed address relation E₀, let Mᵉˣᵗʳᵃ(ω) record an additional identification involving the coded point ω. The unstable set is their union over all infinite addresses; the stable set is its complement in R.
U = ⋃ω Mᵉˣᵗʳᵃ(ω), S = R ∖ U.
The selected-tip layer probes finitely many ultimately periodic addresses. Forbidden prefixes test complete cylinders of addresses. Neither finite survival nor exclusion of a finite subunion establishes full stability. The worker reports its depth and work budget; exact selected-parameter checks are separate.
Green denotes inherited stable coverage. Grey finite exclusions apply only to the test shown. Amber denotes surviving candidates; blue-grey denotes an exhausted budget. Raster pixels represent sampled centres. An algebraic chart stops at its recorded branch isolator.
Origin & implementation
Tip-point sets · Chęciny 2023
Bernat Espigulé presented On the tip-point set M(w) of complex-parametric families of self-similar sets at Complex dynamics: connections to other fields, Chęciny, Poland, 27–31 March 2023.
Conference abstract · Algorithms, evidence and source record
This edition recovers the two earlier family registries, the inverse-search approach in the collinear Canvas explorer, and cached rendering in the 4D atlas. Geometric previews update while the locator moves; parameter fields refine in cancellable workers. A work limit never discards a branch and then declares escape.
Canonical reciprocal coordinates satisfy |c₁| > 1 and |c₂| > 1. The slice coordinate is c = 1/q for zippers and c = 1/a for contact families. Surface area, where reported in source records, is measured in the bounded multiplier chart, not in this projection.