Lecture 7: Linear (Möbius) Transformations

This file formalizes the basic theory of Möbius (linear fractional) transformations and their classification into parabolic, strictly hyperbolic, and elliptic types, following Definition 1 of Lecture 7.

A Möbius transformation is a map of the form S(z) = (az + b)/(cz + d) where a, b, c, d ∈ ℂ and ad - bc ≠ 0.

Möbius transformations act on the Riemann sphere ℂ ∪ {∞} (the extended complex plane). In particular, S(∞) = a/c when c ≠ 0 and S(∞) = ∞ when c = 0. Thus ∞ is a fixed point of S if and only if c = 0.

The normal form (equation (12), p. 86): when S has two distinct fixed points α and β on the Riemann sphere, one can write (S(z) - α)/(S(z) - β) = k · (z - α)/(z - β) where k is the multiplier of S.

When one of the fixed points is ∞ (i.e., c = 0), the normal form reduces to S(z) - α = k · (z - α) with k = a/d.

Classification (Definition 1, Lecture 7)

• Parabolic: S is the identity or has exactly one fixed point on ℂ ∪ {∞}.
• Strictly hyperbolic: the multiplier k is a positive real number with k ≠ 1.
• Elliptic: |k| = 1 and S is not the identity.

structure MoebiusTrans : Type

A Möbius transformation (linear fractional transformation) is a map S(z) = (a·z + b) / (c·z + d) with the non-degeneracy condition a·d - b·c ≠ 0. See Lecture 7, Text pp. 80-89.

• a : ℂ

  The coefficient of z in the numerator.

• b : ℂ

  The constant term in the numerator.

• c : ℂ

  The coefficient of z in the denominator.

• d : ℂ

  The constant term in the denominator.

• det_ne_zero : self.a * self.d - self.b * self.c ≠ 0

  The non-degeneracy condition: ad - bc ≠ 0.

noncomputable def MoebiusTrans.eval (S : MoebiusTrans) (z : ℂ) : ℂ

Evaluate the Möbius transformation at a point z, giving (a·z + b) / (c·z + d).

@[simp]

theorem MoebiusTrans.eval_def (S : MoebiusTrans) (z : ℂ) : S.eval z = (S.a * z + S.b) / (S.c * z + S.d)

def MoebiusTrans.IsFixedPoint (S : MoebiusTrans) (z : ℂ) : Prop

A complex number z is a finite fixed point of S if c·z + d ≠ 0 and S(z) = z.

def MoebiusTrans.fixedPoints (S : MoebiusTrans) : Set ℂ

The set of (finite) fixed points of a Möbius transformation.

def MoebiusTrans.IsFixedPointAtInfty (S : MoebiusTrans) : Prop

A Möbius transformation has a fixed point at ∞ when c = 0. This captures the action on the Riemann sphere: S(∞) = a/c when c ≠ 0, and S(∞) = ∞ when c = 0.

def MoebiusTrans.IsIdentity (S : MoebiusTrans) : Prop

A Möbius transformation is the identity if S(z) = z for all z in its domain (i.e., for all z with c·z + d ≠ 0).

def MoebiusTrans.HasExactlyOneExtFixedPoint (S : MoebiusTrans) : Prop

The total number of extended fixed points (on ℂ ∪ {∞}) is exactly one: either ∞ is the unique fixed point (c = 0 and no finite fixed points), or there is a unique finite fixed point and ∞ is not fixed (c ≠ 0).

noncomputable def MoebiusTrans.multiplier (S : MoebiusTrans) (α β : ℂ) : ℂ

The multiplier of a Möbius transformation S with respect to two finite fixed points α and β. From the normal form (equation (12), p. 86): (S(z) - α) / (S(z) - β) = k · (z - α) / (z - β), one derives k = (a - α·c) / (a - β·c).

noncomputable def MoebiusTrans.multiplierAtInfty (S : MoebiusTrans) : ℂ

The multiplier at ∞: when one fixed point is a finite point α and the other is ∞ (i.e., c = 0), the normal form S(z) - α = k · (z - α) gives k = a/d.

Definition 1: Classification of Möbius Transformations

These definitions act on the Riemann sphere ℂ ∪ {∞}, accounting for the possibility that ∞ is a fixed point (when c = 0).

def MoebiusTrans.IsParabolic (S : MoebiusTrans) : Prop

Definition 1, Lecture 7. A Möbius transformation S is parabolic if either it is the identity or it has exactly one fixed point on the Riemann sphere ℂ ∪ {∞}.

def MoebiusTrans.IsStrictlyHyperbolic (S : MoebiusTrans) : Prop

Definition 1, Lecture 7. A Möbius transformation S is strictly hyperbolic if it has two distinct fixed points on the Riemann sphere ℂ ∪ {∞} and the multiplier k in the normal form (12) satisfies k > 0 (positive real) and k ≠ 1.

• Case 1 (two finite fixed points): α, β ∈ ℂ with α ≠ β, multiplier k = (a - α·c)/(a - β·c).
• Case 2 (one finite, one at ∞): α ∈ ℂ and ∞ (when c = 0), multiplier k = a/d.

def MoebiusTrans.IsElliptic (S : MoebiusTrans) : Prop

Definition 1, Lecture 7. A Möbius transformation S is elliptic if |k| = 1 in the normal form (12) (equivalently, |k|² = 1) but S is not the identity. This requires two distinct fixed points on ℂ ∪ {∞}.

• Case 1 (two finite fixed points): |multiplier α β|² = 1.
• Case 2 (one finite, one at ∞): |a/d|² = 1.