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!