The strong closing lemma and Hamiltonian pseudo-rotations (Q6617229)

This paper establishes a closing lemma for a large class of Hamiltonian diffeomorphisms. It follows the work by \textit{K. Irie} [J. Fixed Point Theory Appl. 26, Paper No. 15, 23 p. (2024; Zbl 1541.53101)], who formulated a strong closing property for contact forms.\N\NThe setting is as follows: \((M, \omega)\) is a closed and connected symplectic manifold and \(\mathrm{Ham}(M, \omega)\) is its group of Hamiltonian diffeomorphisms. This group admits a bi-invariant distance metric \(\gamma\) (called the spectral metric). It is described, for example, in [\textit{Y.-G. Oh}, Prog. Math. 232, 525--570 (2005; Zbl 1084.53076)]. A map \(\psi \in \mathrm{Ham}(M,\omega))\) is called \(\gamma\)-rigid if there is a sequence of integers \(k_i \rightarrow \infty\) such that \(\psi^{k_i} \rightarrow \mathrm{Id}\) in the metric \(\gamma\).\N\NExamples of \(\gamma\)-rigid maps include rotations and some classes of pseudo-rotations.\N\NThe main result says that if \(\psi\) is a \(\gamma\)-rigid Hamiltonian diffeomorphism, then for every non-zero possibly time-dependent Hamiltonian \(G \geq 0\) (or \(G \leq 0\)) supported away from periodic points of \(\psi\), the composition \(\varphi_G\psi\) has a periodic orbit passing through the support of \(G\), where \(\varphi_G\) if the time-one map of the flow of the Hamiltonian \(G\). In particular, every \(\gamma\)-rigid Hamiltonian diffeomorphism has the strong closing property.\N\NThe authors note that the conclusion does not hold without the assumption that \(G \geq 0\) (or \(G \leq 0\)).\N\NThe proof of the main theorem relies on use of Hamiltonian Floer theory for arbitrary closed symplectic manifolds.