SYMMETRIES OF CENTER SINGULARITIES OF PLANE VECTOR FIELDS

Let D-2 subset of R-2 be a closed unit 2-disk centered at the origin O is an element of R-2 and let F be a smooth vector field such that O is a unique singular point of F and all other orbits of F are simple closed curves wrapping once around O: Thus, topologically O is a "center" singularity. Let theta: D-2\ {0} -> (0,+infinity) be the function associating with each z not equal O its period with respect to F: In general, such a function cannot be even continuously defined at O: Let also D+ (F) be the group of diffeomorphisms of D-2 that preserve orientation and leave invariant each orbit of F: It is proved that theta smoothly extends to all of D-2 if and only if the 1-jet of F at O is a "rotation," i. e., j(1)F(O) = -y partial derivative/partial derivative x + x partial derivative/partial derivative y. Then D+(F) is homotopy equivalent to a circle.