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### Open Access Journals

DCDS

We prove that any perturbation of the symplectic part of the derivative of a Poisson diffeomorphism can be realized as the derivative of a $C^1$-close Poisson diffeomorphism.
We also show that a similar property holds for the Poincaré map of a Hamiltonian on a Poisson manifold.
These results are the conservative counterparts of the Franks lemma, a perturbation tool used in several contexts most notably in the theory of smooth dynamical systems.

DCDS

In this paper we give a new proof of the local analytic
linearization of flows on T

^{2}with a Brjuno rotation number, using renormalization techniques.
DCDS

We introduce a renormalization group framework for the study
of quasiperiodic skew flows on Lie groups of real or complex
$n\times n$ matrices, for arbitrary
Diophantine frequency vectors in $R^{d}$
and dimensions $d,n$.
In cases where the Lie algebra component of the vector field is small,
it is shown that there exists an analytic manifold
of reducible skew systems, for each Diophantine frequency vector.
More general near-linear flows are mapped to this case
by increasing the dimension of the torus.
This strategy is applied for the group of unimodular
$2\times 2$ matrices, where the stable manifold
is identified with the set of skew systems having a fixed
fibered rotation number.
Our results apply to vector fields of class C

^{γ}, with $\gamma$ depending on the number of independent frequencies, and on the Diophantine exponent.## Year of publication

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