Dynamical convexity and closed orbits on symmetric spheres

Our main theme here is the dynamics of Reeb flows with symmetries on standard contact sphere. We introduce notion strong dynamical convexity for forms invariant under a group action, supporting structure, and prove that in dimension 2n+1 any such form satisfying condition slightly weaker than has at least n+1 simple closed orbits. For antipodal symmetry, we consequence ordinary convexity. In 5 or greater, construct examples antipodally symmetric, dynamically convex which are not strongly convex, thus contactomorphic to ones via contactomorphism commuting map. Finally, relax this furnishing nonempty C1-interior.