Holomorphic Dynamics and Mobius Transformations

MTH3000 Final Report - Additional videos shown below report.

Example Let $\varphi : \mathbb{D} \mapsto \mathbb{D}$ be defined by

\[\varphi(z) = iz\]

$\varphi$ is elliptic since it has one fixed point in $\mathbb{D}$.

Example Let $\varphi : \mathbb{H} \mapsto \mathbb{H}$ be defined by

\[\varphi(z) = 2z\]

$\varphi$ is parabolic since $(\text{tr} \varphi)^2 = 9$.

Example Let $\tau = 1$ and $\alpha = 1$. Iterating $\varphi_t$, on the upper half plane the points are translated along horoballs at the fixed point, moving away from one side of the fixed point towards the other.

Let $\tau = i$ and $\alpha = 1$. Iterating $\varphi_t$, on the upper half plane the points are translated along horoballs at the fixed point. Since the fixed point is at $\infty$, the horoballs are horizontal lines, so the points are simply translated in the plane.

Example Let $\tau = 0.5$ and $\omega = \frac{\pi}{2}$. Iterating $\varphi_t$, the points orbit around the fixed point, $0.5$.

Example Let $\sigma = 1$, $\tau = i$ and $\alpha = 1$. Iterating $\varphi_t$, the points move away from one fixed points towards the other along geodesics, similarly to the behaviour of Hyperbolic Möbius transformations

Let $\sigma = 1$, $\tau = -1$ and $\alpha = 1$.