General affine group is not conjugacy-closed in self-homeomorphism group

Statement

The general affine group $GA(n,\R)$ of all invertible affine transformations over the field of real numbers, is not conjugacy-closed in the self-homeomorphism group of $\R^n$. In other words, there are affine transformations which are not conjugate in $GA(n,\R)$ but are n

In particular, this also yields that the self-diffeomorphism group of $\R^n$, is not conjugacy-closed in the self-homeomorphism group, because general affine group is conjugacy-closed in self-diffeomorphism group.

Proof

Put $n=1$. Then the transformations $x \mapsto ax$ and $x \mapsto bx$, are conjugate in the self-homeomorphism group if $a,b$ are both between $0$ and $1$. The conjugating homeomorphism is the map:

$x \mapsto x|x|^{\log b / \log a - 1}$

and is defined to be $0$ at $0$.

Note that this conjugating homeomorphism is not a diffeomorphism.

This same example can be used for the case $n > 1$, by taking scalar matrices for $a$ and $b$.