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I guess my question is very vague. Let $\phi: \mathbb{R}^2 \to \mathbb{R}^2$ be an orientation-preserving diffeomorphism that fixes the origin. Does $\phi$ take any sort of "normal form" if we allow any conjugation by another orientation-preserving, origin-fixing diffeomorphism of $\mathbb{R}^2$?

Or rather, does anyone knows some references that study conjugacy classes of self-diffeomorphisms of $\mathbb{R}^2$?

Thanks a lot!

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    $\begingroup$ The study of self-diffeomorphisms of $\mathbb{R}^2$ up to conjugacy is part of the theory of smooth dynamical systems. A lot is known, but it is also known that there are many complicated phenomena, starting with the classification of stable and unstable manifolds and the nonwandering set, periodic points, etc. There is no known 'normal form', as far as I know. $\endgroup$
    – Robert Bryant
    Commented Sep 5, 2014 at 12:23
  • $\begingroup$ Thanks Robert! The classification of stable and unstable manifolds part might be interesting to me. Any references like books or surveys would be greatly appreciated, as I cannot find much by just google search.. $\endgroup$
    – user57879
    Commented Sep 5, 2014 at 12:37
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    $\begingroup$ A good early reference is Smale, S. (1967), "Differentiable Dynamical Systems", Bull. Amer. Math. Soc. 73: 747–817, which is available online for free. I don't do dynamical systems, so I don't know the best modern references. You should try looking at the Wikipedia page for "Axiom A" and follow some of those references for background reading. Good luck. $\endgroup$
    – Robert Bryant
    Commented Sep 5, 2014 at 12:54
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    $\begingroup$ Up to isotopy the classification is the same as for affine transformations, see front.math.ucdavis.edu/0109.5183 for the homotopy classification of diffeomorphism groups of open surfaces. $\endgroup$
    – Igor Belegradek
    Commented Sep 5, 2014 at 21:36
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    $\begingroup$ @IgorBelegradek the link in your comment is broken, here's a replacement: arxiv.org/abs/math/0109183 $\endgroup$
    – David Roberts ♦
    Commented Mar 29, 2022 at 1:12

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This is hopeless. Even if you allow conjugation by homeomorphisms (that is topological classification), and restrict your class of diffeomorphisms to those which have the form $z\mapsto \lambda z+\phi(z)$ near the origin, with complex analytic $\phi$, it is very complicated, see, for example, MR1241470.

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