Stable conjugation invariants of Hamiltonian systems on two-dimensional surfaces
Let \(M\) be a connected orientable two-dimensional surface with or without boundary, \(F:M\rightarrow {\mathbb R}\) a Morse function which is constant and without critical points on any component of the boundary, and \(\omega\) a symplectic structure on \(M\). Two Morse functions \(F\) and \(F'\) are levelwise equivalent if there exists an orientation preserving diffeomorphism which transforms connected components of the level sets of \(F\) into connected components of the level sets of \(F'\). If in addition the set \(\{ F<c\}\) is mapped into the set \(\{ F'<c'\}\), then \(F\) and \(F'\) are levelwise framed-equivalent. Consider the Hamiltonian system with Hamiltonian \(F\) on \(M\). For a critical value \(c\) of \(F\) suppose that the set \(K=F^{-1}(c)\) is connected and contains only saddle critical points. Select \(\varepsilon >0\) so small that there are no other critical values of \(F\) in \([c-\varepsilon ,c+\varepsilon ]\). The class of levelwise equivalence (resp. of levelwise framed equivalence) of \(F\) in the neighbourhood \(P=F^{-1}[c-\varepsilon ,c+\varepsilon ]\) of its critical level \(K\) is completely determined by the class of topological equivalence of the pair \((P,K)\) (resp. \((P,K_+)\)) considered together with the orientation of the surface \(P\) (resp. with the orientations of \(P\) and of the graph \(K\)). The pairs \((P,K)\) and \((P,K_+)\) are called the saddle atom and the saddle \(f\)-atom respectively, corresponding to the function \(F\) on the surface \(P\). Two Hamiltonian systems \(v_1\) and \(v_2\) on an \(f\)-atom \((P,K_+)\) are stably not conjugate near saddle critical levels, if they are not conjugate in any arbitrarily small neighbourhoods \(U_1\) and \(U_2\) of the saddle levels, and for any small perturbations of the functions \(F_i\) and symplectic structures \(\omega _i\), \(i=1,2\), the perturbed systems remain nonconjugate in any domains \(\widetilde{U}_1\) and \(\widetilde{U}_2\), \(U_i\subset \widetilde{U}_i\). The paper announces several theorems on two Hamiltonian systems being stably not conjugate.
stably not conjugate Hamiltonian systems
scientific article; zbMATH DE number 1557426