Let ρ: R²→ R² be a nonidentity rotation about the point P. Describe geometrically the ρ -orbits on the set P of all points of R². Does this explain why orbits are called orbits?
Oh, c'mon! This is as easy as a problem can get! Look at a simple example. Suppose p rotates points around the origin by angle 45 degrees. What is p((1,0))? what is p(p((1,0)))? What is p(p(p((1,0))))? Do you see the point?