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A couple of days ago our professor reviewed some of the exercises we had to do and one of them involved giving an example of a conjugacy class in a group.

Someone gave an example that involved invertible functions from $ C^\infty (\mathbb{R},\mathbb{R}^n)$ and a map $h\mapsto f h f^{-1}$ (I can't recall exactly what the example was, since I hadn't time to look properly at it. Maybe you can helpt me figure out what it was ?)

This example totally freaked the professor out (in a positive way) because he said that finding the conjugacy class for this example was something extremely difficult and still subject to open research, being called the "conjugacy problem for differentiable maps" (if I remember well) and that all kind different things from different areas of mathematics (like approximating reals with rationals) are invovled in establishing the conjugacy class for these functions.

But after a search in google, I couldn't find much information on the so-called "conjugacy problem for differentiable maps". Could you tell me what he was talking about and why this stuff is so important ?

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Perhaps you could stop by your professor's office to talk about it? I am sure he would be happy to discuss it. –  Grumpy Parsnip Nov 4 '11 at 16:53
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Is $n=1$? Otherwise, invertible functions in $C^{\infty}(\mathbb{R},\mathbb{R}^n)$ would not form a group under composition. –  Arturo Magidin Nov 4 '11 at 17:04
    
I don't think $C^\infty(\mathbb R, \mathbb R^n)$ contains any invertible functions when $n\ne 1$. –  Henning Makholm Nov 4 '11 at 18:43
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I think you'll get more results if you search for "topological conjugacy problem," or "conjugacy problem for diffeomorphisms." Why not have a look, and then let us know what you have found?

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