Symmetric periodic Reeb orbits on the sphere (Q6597547)

The authors consider a refinement of an old conjecture from Hamiltonian dynamics. The long-standing conjecture suggests that every contact form on the standard contact sphere \(S^{2n+1}\) has at least \(n + 1\) simple periodic Reeb orbits.\N\NOn \(S^{2n+1}\) with the Liouville form \[\lambda = \frac{1}{2} \sum_{i=1}^{n+1} q_i dp_i - p_i dq_i,\] the standard contact structure is \(\xi =\) ker \(\lambda_{S^{2n+1}}\) and the contact form on this structure is a 1-form \(\alpha = f\lambda|_{S^{2n+1}}\) where \(f:S^{2n+1} \rightarrow \mathbb{R}\) is a positive function. With \(\alpha\) is the associated Reeb vector field \(R_\alpha\) determined by the equations \(i_{R_\alpha} d\alpha = 0\) and \(\alpha(R_\alpha) = 1\).\N\NThe authors' main theorem is that if \(\alpha\) is any contact form on \(S^{2n+1}\) invariant under a certain \(\mathbb{Z}_p\) - action on \(S^{2n+1}\), then \(\alpha\) has at least one symmetric closed orbit. The \(\mathbb{Z}_p\) action is defined by \[\psi(z_0, ...,z_n) = \left(e^\frac{2\pi i l_0}{p} z_0, e^\frac{2\pi i l_1}{p} z_1, ...,e^\frac{2\pi i l_n}{p} z_n\right),\] where \(l_0, l_1,\dots,l_n\) are the integral weights of the action. It is assumed that the weights are coprime with \(p\). Then the action is free, and the quotient of \(S^{2n+1}\) under the action of \(\mathbb{Z}_p\) is a lens space.\N\NThe authors also prove that there are at least two symmetric closed orbits when the contact form is dynamically convex.