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The rough idea I had was that RP^2 is the collection of all lines in R^3 passing through the origin, so if we rotate each line by some angle, then the origin is a fixed point and all other points have periodic orbits.

Don't know how to formally proceed though.

Thanks for the help!

Akshat Das
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  • Note that there is no point in the projective plane which corresponds naturally to the origin in $\mathbb R^3$. – Blazej Mar 25 '18 at 23:09

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You can use the fact that the projective plane is equivalent to a sphere with antipodal points identified. Thus it is sufficient to find a flow on $S^2$ which is invariant under reflection (so that it lifts to projective space unambigously) and has two (antipodal) fixed points and all other orbits periodic. Can you do it? Hint: there is a particularly simple example with nice geometric interpretation.

Blazej
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  • So if we consider the rotation matrix in R^3 then all points on S^2 except the north pole have periodic orbits whereas the north pole remains constant right? – Akshat Das Mar 26 '18 at 00:01
  • Right, but also the south pole is a fixed point. However in the projective space north and south pole are identified as one point. – Blazej Mar 26 '18 at 20:15