Möbius Transformations

Möbius Transformations (on $\mathbb{C}$)

A möbius transformation is a function $f:\mathbb{C}\to \mathbb{C}$ with $f(z)=\frac{az+b}{cz+d}$ with $a,b,c,d\in \mathbb{C}$ and $ad-bc\ne 0$. Set $f(\infty )=\frac{a}{c}$ when $c\ne 0$, $f(\infty )=\infty$ when $c=0$, we get extended definition on $\hat{\mathbb{C}}$.

Möbius Transformations

The set of Möbius transformations is given by $\mathrm{M}\ddot{\mathrm{o}}\mathrm{b}=\{{\lambda }_X:\hat{\mathbb{C}}\to \hat{\mathbb{C}}\mid X\in {\mathrm{GL}}_2(\mathbb{C})\}$where for any $X=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\mathrm{GL}}_2(\text{C})$, ${\lambda }_X(z)=\{\begin{array}{ll}\frac{az+b}{cz+d}& z\ne \infty ,-d/c\\ \infty & z=-d/c\\ a/c& z=\infty \end{array}$where ${\mathrm{GL}}_2(\text{C})$ is the general linear group over $\text{C}$. We will often write ${\lambda }_X(z)=\frac{az+b}{cz+d}$, since the other two cases are compatible with the limits of the fraction.

Note $A=\left(\begin{array}{cc}\lambda & 0\\ 0& \lambda \end{array}\right)⟹f_A=id$ Aside: { Mobius transforms } $\cong PG{L}_2(\mathbb{C})=G{L}_2(\mathbb{C})/\left(\begin{array}{cc}\lambda & 0\\ 0& \lambda \end{array}\right)$ $PG{L}_2(\mathbb{R})$ used in Hyperbolic Geometry $PG{L}_2(\mathbb{Z})$ used in Number Theory

Dynamical Behaviours

A Möbius transformation $f(z)=\frac{az+b}{cz+d}$ is naturally associated with a matrix $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in G{L}_2(\mathbb{C})$. To classify the dynamics algebraically, we must first normalize the matrix so that its determinant is $1$. We map the transformation to the Special Linear Group $S{L}_2(\mathbb{C})$ by dividing by the square root of the determinant $\Delta =ad-bc$: $A_{\text{norm}}=\frac{1}{\sqrt{\Delta }}\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ The fundamental invariant under conjugation in $S{L}_2(\mathbb{C})$ is the trace squared, denoted as ${\tau }^2$: ${\tau }^2=\text{tr}(A_{\text{norm}}{)}^2=\frac{(a+d{)}^2}{ad-bc}$ Because conjugate dynamical systems (similar matrices) share the same trace, ${\tau }^2$ uniquely dictates the geometric behavior of the transformation on the Riemann sphere $\hat{\mathbb{C}}$.

2. The Classification

Based on the value of ${\tau }^2$, any non-identity Möbius transformation falls into one of the following distinct dynamical categories:

Parabolic

• Algebraic Condition: ${\tau }^2=4$
• Fixed Points: Exactly $1$ fixed point in $\hat{\mathbb{C}}$.
• Conjugate Standard Form: $f(z)=z+1$ (associated with $\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$)
• Dynamical Behavior: Pure translation. All orbits flow along mutually tangent circles toward the single fixed point.

Elliptic

• Algebraic Condition: $0\le {\tau }^2<4$ (where ${\tau }^2\in \mathbb{R}$)
• Fixed Points: Exactly $2$ distinct fixed points.
• Conjugate Standard Form: $f(z)=e^{i\theta }z$ for $\theta \in \mathbb{R}\setminus \{2\pi k\}$ (associated with $\left(\begin{array}{cc}{e}^{i\theta /2}& 0\\ 0& e^{-i\theta /2}\end{array}\right)$)
• Dynamical Behavior: Pure rotation. Orbits move along invariant circles surrounding the two fixed points, with no points converging to either.

Hyperbolic

• Algebraic Condition: ${\tau }^2>4$ (where ${\tau }^2\in \mathbb{R}$)
• Fixed Points: Exactly $2$ distinct fixed points ($1$ attractor, $1$ repeller).
• Conjugate Standard Form: $f(z)=\lambda z$ for $\lambda \in {\mathbb{R}}^{+}\setminus \{1\}$ (associated with $\left(\begin{array}{cc}\sqrt{\lambda }& 0\\ 0& 1/\sqrt{\lambda }\end{array}\right)$)
• Dynamical Behavior: Pure dilation. Points are repelled from the source (repeller) and move along circular arcs directly toward the sink (attractor).

Loxodromic (General Case)

• Algebraic Condition: ${\tau }^2\notin [0,\infty )$ (i.e., ${\tau }^2$ is strictly complex or strictly negative).
• Fixed Points: Exactly $2$ distinct fixed points.
• Conjugate Standard Form: $f(z)=\lambda z$ for $\lambda \in \mathbb{C}$, where $|\lambda |\ne 1$.
• Dynamical Behavior: A combination of dilation and rotation. Orbits form logarithmic spirals, spiraling out of the repelling fixed point and into the attracting fixed point.

(Note: Hyperbolic transformations are technically a special, strictly real case of Loxodromic transformations).

Hyperbolic Plane

${\mathbb{H}}^2$

${\mathbb{H}}^2=\{(x,y)\in {\mathbb{R}}^2|y>0\}=\{z\in \mathbb{C}|\text{Im}(z)>0\}$

Metric on ${\mathbb{H}}^2$

Consider a curve $\gamma$ from $(x(a),y(a))$ to $(x(b),y(b))$, parametrized by a differentiable function: $t\to (x(t),y(t))$ for $a\le t\le b$. The hyperbolic length of this curve is defined by the integral: ${\ell }_{hyp}(\gamma )={\int }_a^b\frac{\sqrt{x^{\prime }(t{)}^2+y^{\prime }(t{)}^2}}{y(t)}dt$ Given two points $P,Q\in {\mathbb{H}}^2$, the hyperbolic distance $d_{hyp}(P,Q)$ is defined as the infimum of the lengths of all possible paths connecting them: $d_{hyp}(P,Q)=\inf \{{\ell }_{hyp}(\gamma )\mid \gamma \text{ is a piecewise differentiable curve from }P\text{ to }Q\}$

Theorem

Möbius Transformations are isometries of ${\mathbb{H}}^2$.