Possibilities of an action of $S^1$ on a disk.

I’m dealing with actions of the circle over differentiable manifolds. In the book I’m reading, they use the fact that an action of $S^1$ over a disk has to be equivalent (there has to exist an equivariant diffeomorphism) to a rotation. Can someone give me a hint to prove this? Or maybe a reference where I can consult this result? It seems to be a “well known fact” but I haven’t been able to find a place where they prove it.

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Your question is a little imprecise but I’ll take it to be the assertion that a smooth action of $S^1$ on $D^n$, i.e. a smooth homomorphism $S^1 \to Diff(D^n)$ is conjugate via a diffeomorphism of $D^n$ to a homomorphism $S^1 \to SO_n$.

First off, the generator of the motion is a vector field on $D^n$ which is tangent to $\partial D^n = S^{n-1}$. Since its tangent on the boundary, you can perturb the vector field near the boundary to be outward-pointing. Poincare-Hopf kicks in and tells you this vector field needs to have a zero on the interior, so your original vector field has a zero on the interior.

So in the orbit decomposition of $D^n$ you have a non-empty fixed point set. So all we need to do is show the fixed point set is isotopic to a linear subspace — once you have that you know that the disc $D^n$ is just an $S^1$-equivariant tubular neighbourhood of that fixed point set, and so its characterized by its behaviour near the fixed point set, which is linear.

Hmm, this part seems likely to not be true. Apparently there are actions of circles on discs with precisely two fixed points in the interior. Ref I don’t have access to the paper from home so I haven’t looked at anything other than the first page, but this appears to be a non-linear action. Perhaps in the reference you’re referring to they’re content with a local result rather than a global result? You might want to check the wording carefully. Locally it’s certainly true by the above argument.