Mirror-symmetric complex trees
Bernat Espigulé · The 2013 contact families, reciprocal coordinates and their connection to the four-dimensional direct locus.
From mirror symmetry to the direct connectedness locus
The binary mirror slice has a = reiθ and b = re−iθ, with 0 < r < 1. Both maps preserve orientation. It lies in the direct locus Mff, where f₀(z) = −1 + az and f₁(z) = 1 + bz. In the 4D chart this is w = ½, α = θ, γ = 0, ρ = 1/r.
A contact equation specifies an identification of infinite addresses. A marked stable family also preserves the complete coding relation on a stated domain. The contact atlas distinguishes the equation, its certified stable subset and the numerical drawing. Families with more than two maps have 2n real multiplier coordinates.
The historical canopy uses Fⱼ(z) = qⱼ(1 + z); its finite branch skeleton is a separate drawing. For a ≠ b, translation by 1 followed by H(z) = ((2 − a − b)z − 2)/(b − a) carries the binary canopy to the anchored maps above. On this page the exterior mirror coordinate is c = 1/conj(q), so (c₀,c₁) = (conj(c),c). Notation and coordinate conventions.
Explore the continuous arity-transition tracksOpen the transition panel below
Read the geometric specimen
Transition curves between branching families
Follow a transition between adjacent contact formulas as ν varies. Circles mark the actual integer-arity fans. Change coordinate charts without changing the selected parameter.