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Classification of nonresonant Poisson structures. (English) Zbl 0940.37022
We study Poisson structures on an $$(n+1)$$-manifold which vanish at a point $$p$$. We say that the Poisson structure is nonresonant at $$p$$ if there is at least one Hamiltonian vector field $$X$$ such that the linear part of $$X$$ at $$p$$ has eigenvalues $$l_1,\dots,l_n$$ without non trivial “resonance relations” $$l_i+l_j=m_1l_1+\dots+m_nl_n$$ ($$m_j$$ positive integers). We give formal, smooth and analytic normal forms for these Poisson structures near $$p$$. These normal forms are quadratic and they generalise some previous linearisation results.

MSC:
 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion 53D17 Poisson manifolds; Poisson groupoids and algebroids
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