Dynamical implications of convexity beyond dynamical convexity

We establish sharp dynamical implications of convexity on symmetric spheres that do not follow from dynamical convexity. It allows us to show the existence of elliptic and non-hyperbolic periodic orbits and to furnish new examples of dynamically convex contact forms, in any dimension, 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. We also show the multiplicity of symmetric non-hyperbolic and symmetric (not necessarily non-hyperbolic) closed Reeb orbits under suitable pinching conditions.

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