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On the topology of an integrable Hamiltonian system in the neighborhood of a degenerate one-dimensional trajectory. (Russian) Zbl 0855.58024

Let \(s\text{grad } h\) be an integrable Hamiltonian system on a compact 4-dimensional manifold \(M^4\) with a symplectic structure. Let \(f\) be its first integral, a.e. independent of \(h\), \(Q^3= \{h(x)= E\}\) an isoenergetic surface, \(F(x)=(h(x), f(x))\) a momentum map, and let \(K\) be the set of singular points of \(F\) and \(\Sigma= F(K)\) the bifurcation diagram. Let \(A\) be a point of a singular one-dimensional orbit \(O\) and \(U\) some neighbourhood of \(A\) in the two-dimensional manifold transversal to \(O\) in \(Q\). Denote by \(\Phi\) the Poincaré return map on \(U\). Then using the classification of \(d_A\Phi\) \((\text{det } d_A \Phi=1)\) and some invariance one obtains (under some mild assumption) that \(d_A \Phi^q=\text{id}\) with \(q\) less or equal to the degree of \(f\) at \(A\). Further, in some cases there exists a unique power series \(H\) so that \(\Phi\) is a time 1 map of the vector field \(s\text{grad } H\).

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
54H20 Topological dynamics (MSC2010)
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