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

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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.


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.