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 $$b$$.