Linearization of smooth integrable systems

I’m writing down here the ideas for proving that smooth nondegenerate integrable dynamical systems are smoothly linearizable. The analytic case can be proved using analytic torus actions (my paper about that will appear in Ergodic Th Dyn Sys). I think the smooth case is also true, but the proof is much more complicated than the analytic case, because we don’t have a smooth torus action of full dimension in general due to non-elliptic components. We need a combination of other additional techniques.

Torus action

A torus action still exists. It’s dimension is the real toric degree (not the full toric degree). We can still assume that this real torus action is linear and is part of the system, and that the remaining components of the system are jointly hyperbolic.

 

 

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