Our main theme here is the dynamics of Reeb flows with symmetries on the standard contact sphere. We introduce the notion of strong dynamical convexity for contact forms invariant under a group action, supporting the standard contact structure, and prove that in dimension any such contact form satisfying a condition slightly weaker than strong dynamical convexity has at least simple closed Reeb orbits. For contact forms with antipodal symmetry, we prove that strong dynamical convexity is a consequence of ordinary convexity. In dimension 5 or greater, we construct examples of antipodally symmetric, dynamically convex contact forms which are not strongly dynamically convex, and thus not contactomorphic to convex ones via a contactomorphism commuting with the antipodal map. Finally, we relax this condition on the contactomorphism furnishing a condition that has nonempty -interior.
"Dynamical convexity and closed orbits on symmetric spheres." Duke Math. J. 170 (6) 1201 - 1250, 15 April 2021. https://doi.org/10.1215/00127094-2020-0097