TITLE: circle action on sphere
QUESTION [5 upvotes]: surely $S^1$ can act on $S^n$ as a rotation.I want to know if there is some other way that a circle can act on sphere.

REPLY [14 votes]: Classification of circle actions on (standard and on exotic) spheres is a classical activity in transformation groups, see e.g. the article of Schultz in the collection "Group actions on manifolds", proceedings from 1983 conference, or earlier account in Bredon's book "Introduction to compact transformation groups". 
One standard example is this. Take a smooth compact contractible $(n+1)$-manifold $C$ and consider $C\times D^k$ where circle acts trivially on $C$ and linearly
on the $k$-disk $D^k$, $k>0$. Now if $n+k>4$, the boundary of $C\times D^k$ is diffeomorphic to the standard sphere (after the corners of $C\times D^k$ are rounded). But the fixed point set of the action is the original homology sphere that bounds $C$. I think it is clear that the fixed point sets of linear actions are standard spheres.
Incidentally, in dimensions $k>4$ any homology $k$ sphere bounds a contractible manifold after possibly changing its smooth structure; also any homology $4$-sphere bounds a smooth contractible manifold, while any homology $3$-sphere bounds a topological contractible manifold, but not necessarily smooth ones. References to the above results can by found e.g. in my paper pp.8-9.