Let $\Omega$ be $(\mathbb{R}^-\cup{0})^c$. Find an automorphism that has $1$ as a fixed point and it is not the identity.
Could an inversion of the plane across the $xy$ axis be an automorphism that satisfies the condition? Since the set seems to be the upper left quadrant I'm thinking of having the function defined as $\phi (z)=1/z$.