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Unique resonant normal forms for area-preserving maps at an elliptic fixed point. (English) Zbl 1185.37142
From the abstract: We study possible simplifications of normal forms for area-preserving maps near a resonant elliptic fixed point. In the generic case we prove that at all orders the Takens normal form vector field can be transformed to the particularly simple form described by the formal interpolating Hamiltonian $H=I^2 A(I) +I^{n/2} B(I) \cos n\varphi.$ The form of formal series $$A$$ and $$B$$ depends on the order of the resonance $$n$$. For each $$n\geq 3$$ we establish which terms of the series can be eliminated by a canonical substitution and derive a unique normal form which provides a full set of formal invariants with respect to canonical changes in coordinates.
We extend these results to families of area-preserving maps. Then the formal interpolating Hamiltonian takes a form similar to the case of an individual map but involves formal power series in action $$I$$ and the parameter.

##### MSC:
 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion 37G05 Normal forms for dynamical systems
##### Keywords:
normal forms; area-preserving; resonant
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