We will show sharp dynamical implications of convexity on symmetric spheres that do not follow from dynamical convexity. It allows us to furnish new examples of dynamically convex contact forms that are not equivalent to convex ones via contactomorphisms that preserve the symmetry. Moreover, these examples are $C^1$-stable in the sense that they are actually not equivalent to convex ones via contactomorphisms that are $C^1$-close to those preserving the symmetry. Other applications are the multiplicity of symmetric non-hyperbolic closed Reeb orbits under suitable pinching conditions and the existence of symmetric elliptic periodic Reeb orbits.

This is ongoing joint work with Miguel Abreu.