Continuous orbital invariants of integrable Hamiltonian systemsстатья
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Аннотация:We study integrable Hamiltonian systems with 2 degrees of freedom on regular compact isoenergy 3-manifolds. Such a system is given by a pair (B,F) of a closed 2-form B without zeros and a Bott function F (called the first integral) with d(FB)=0 on a compact 3-manifold endowed with a volume form. We prove that, under some additional assumptions, any continuous orbital invariant of integrable systems can be expressed in terms of local extremes of rotation functions on one-parameter families of invariant tori, provided that the systems admit a cross-section of genus 0. We also prove a similar assertion for systems having at most three singular circles on the same connected component of a singular level of the first integral.