How to look for the invariant point of a complex numbers

To find the invariant point (or fixed point) of a complex number transformation, you set the output of the transformation equal to the input. If a transformation is given by $w = f(z)$, an invariant point $z_0$ satisfies the condition $z_0 = f(z_0)$.

For example, if the transformation is a Möbius transformation $w = \frac{az+b}{cz+d}$, you would set $z = \frac{az+b}{cz+d}$ and solve for $z$. This typically leads to a quadratic equation, yielding one or two invariant points.