ALIBABA 2021 Problem 12

Let $A=(a_{ij})$ be a $5 \times 5$ matrix with $a_{ij}=\min\{i,j\}$ . Suppose $f:\mathbb{R}^5 \to \mathbb{R}^5$ is a smooth map such that $f(\Sigma) \subset \Sigma$ , where $\Sigma=\{x \in \mathbb{R}^5: xAx^T=1\}$ . Denote by $f^{(n)}$ te $n$ -th iterate of $f$ . Prove that there does not exist $N \ge 1$ such that $$\inf_{x \in \Sigma} \| f^{(n)}(x)-x\|>0, \forall n \ge N.$$