MathDB

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)}

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.

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