The fixed points of exact symplectomorphisms
Let \((M, \omega)\) be a closed symplectic manifold. A symplectomorphism \(\Phi\) of \((M, \omega)\) is said to be exact if it can be obtained by integrating a time-dependent Hamiltonian vector field. More precisely, \(\Phi\) will be exact if there exists a time-dependent Hamiltonian \(H: M\times [0, 1]\to \mathbb{R}\) such that, defining the flow \(\Phi_t\) by \[ {d\over dt} \Phi_t= X_{H_t}\circ \Phi_t,\quad \Phi_0= \text{id}_M, \] where \(X_{H_t}\) satisfies \(i(X_{H_t}) \omega= dH_t\), one has \(\Phi= \Phi_1\). In the seventies, Arnold conjectured that any exact symplectomorphism \(\Phi\) must have at least as many fixed points as the minimal number of critical points for a smooth real function on \(M\). Moreover, if all fixed points are degenerate, then the lower bound is given by the minimal number of critical points for a Morse function on \(M\). The aim of this short note is to prove that, if \((M_j, \omega_j)\) is a 4-dimensional connected symplectic manifold with \(c_1(M_j)= 0\), \(j= 1,\dots, k\), and \((P, \omega)\) is a closed symplectic manifold with \(\omega|_{\pi_2(P)}= 0\), then every exact symplectomorphism on \((M_1\times\cdots\times M_k\times P, \omega_1\times\cdots\times \omega_k\times \omega)\) has at least one fixed point.
scientific article; zbMATH DE number 797284