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Monodromy in two degrees of freedom integrable systems. (English) Zbl 0776.58017
Let $$f:M^ 4\to\mathbb{R}^ 2$$ be an integrable Hamiltonian system. Assume three conditions:
1.) The set of critical values of $$f$$ contains an isolated point $$p$$. $$f$$ is singular only at one point $$x_ 0\in f^{-1}(p)$$.
2.) All regular fibers $$f^{-1}(r)$$ for $$r$$ near $$p$$ are diffeomorphic to the 2-torus.
3.) Near $$x_ 0$$, $$f$$ has a certain normal form.
Then for any fixed regular value $$r_ 0$$ near $$p$$ and any simple positively oriented loop around $$p$$ connecting $$r_ 0$$ to itself, the monodromy operator $$H_ 1(f^{-1}(r_ 0);\mathbb{Z})\to H_ 1(f^{- 1}(r_ 0);\mathbb{Z})$$ is shown to be represented by the matrix $${1\;0\choose 1\;1}$$ in some appropriately chosen basis for $$H_ 1(f^{-1}(r_ 0);\mathbb{Z})$$.
Reviewer: C.Bär (Bonn)

##### MSC:
 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
##### Keywords:
Hamiltonian systems; monodromy
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##### References:
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