Many closed symplectic manifolds have infinite Hofer-Zehnder capacity (Q2844735)

Assuming that any Hamiltonian system \(H:M\rightarrow \mathbb{R}\) on a closed symplectic manifold \((X,\omega )\) has a nonconstant periodic orbit, then one can derive that for any Hamiltonian system the set of regular values \(c\) such that the characteristic foliation of \(H^{-1}(c)\) has a closed leaf is dense in the range of \(H\). One says then that the ``dense existence property'' holds in \( (X,\omega )\). There existNEWLINENEWLINECriteria:NEWLINENEWLINE1) If the dense existence property holds, then the corresponding Hofer-Zehnder capacity \(c_{HZ}(X,\omega )\) is finite;NEWLINENEWLINE2) (see [\textit{G. Lu}, Isr. J. Math. 156, 1--63 (2006; Zbl 1133.53059)]) If a closed symplectic manifold \((X,\omega )\) is \(GW_{g}\)-connected, that is the Gromow-Witten invariant NEWLINE\[NEWLINE GW_{g},\quad k+2,\quad A([pt],[pt];\beta _{1},\beta _{2},\dots,\beta _{k};B)\neq 0, NEWLINE\]NEWLINE then it has a finite Hofer-Zehnder capacity \(c_{HZ}(X,\omega )<([\omega ],A)\).NEWLINENEWLINEThe author poses the question about nonconstant periodic orbits on closed symplectic manifolds for which the dense existence property does \textit{not }hold.NEWLINENEWLINEDefinition. A symplectic structure \(\omega \) is said to be aperiodic if there exists a compactly supported smooth function \( H:M\rightarrow \mathbb{R}\) which is not everywhere locally constant, such that the Hamiltonian flow of \(H\) has no nonconstant periodic orbits.NEWLINENEWLINEThe following statement is proved:NEWLINENEWLINEProposition. A symplectic form \(\omega \) on a closed manifold \(X\) is aperiodic iff there is a closed cooriented hypersurface \(Y\subset X\), which violates the nearby existence property with respect to \((X,\omega)\).NEWLINENEWLINEThe following question, still open, is posed:NEWLINENEWLINEQuestion. Is the property of being aperiodic invariant under deformation of the symplectic structure?NEWLINENEWLINEMany interesting examples of symplectic manifolds on which the corresponding Hamiltonian flows have no constant periodic orbits are constructed by means of performing symplectic sums along suitable tori and then perturbing the symplectic structure in such a way that hypersurfaces near the ``neck'' have no closed characteristics. It is also conjectured that any closed symplectic 4-dimensional manifold with the characteristic number \(b^{+}(X)>1\) admits symplectic forms with a similar property.