Let $\rho>1$ and $E_\rho$ be the interior of the ellipse $\Gamma: \frac{1}{2}(\rho e^{it}+\rho^{-1} e^{-it}), \quad t \in[0,2\pi]$. Let $A_\rho = \{z: \rho^{-1} < |z| < \rho\}$.
Consider the Joukowsky map $J(z)=\frac{1}{2}(z+z^{-1})$.
I am reading a book where the author says:
For each $\xi \in E_\rho \backslash[-1,1]$ there are two corresponding values of $z \in A_{\rho}$ which are inverses of one another s.t. $J(z)=\xi$.
Why is the line $[-1,1]$ removed? Why is it special? I guess the same statement holds also for $\xi \in [-1,1]$, because we can write $\xi = \frac{1}{2}( e^{it}+e^{-it})$ then $\xi = J(e^{it})=J(e^{-it})$.
Don't we have $J(A_\rho) = E_\rho$?.