A fixed point theorem says that:
"any continuous mapping of $\mathbb{R}^n$ into a bounded subset of $\mathbb{R}^n$ has a fixed point".
So consider $f: \mathbb{R}^n \rightarrow X \subset \mathbb{R}^n$ where $X$ is bounded. Is true that $f$ has a fixed point even if $X$ is not convex, as in the Brouwer fixed point theorem?
I am wondering about the following example: $X = \{ x \in \mathbb{R}^2 \mid x_1^2 + x_2^2 = 1 \}$ and $f(x) = \left[ \begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right] x \in X$.
Is my example wrong and am I missing the meaning of the above Theorem?
Note that if $X$ is bounded, $X$ is contained in some bounded convex closed ball $B$. Then, since $f_{\mid B}$ sends $B$ to $B$, it must have a fixed point.