Abstract
It is well known that a real analytic symplectic diffeomorphism of the \(2d\)-dimensional disk (\(d\geq 1\)) admitting the origin as a non-resonant elliptic fixed point can be formally conjugated to its Birkhoff Normal Form, a formal power series defining a formal integrable symplectic diffeomorphism at the origin. We prove in this paper that this Birkhoff Normal Form is in general divergent. This solves, in any dimension, the question of determining which of the two alternatives of Pérez-Marco’s theorem (Ann. Math. (2) 157:557–574, 2003) is true and answers a question by H. Eliasson. Our result is a consequence of the fact that when \(d=1\) the convergence of the formal object that is the BNF has strong dynamical consequences on the Lebesgue measure of the set of invariant circles in arbitrarily small neighborhoods of the origin. Our proof, as well as our results, extend to the case of real analytic diffeomorphisms of the annulus admitting a Diophantine invariant torus.
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To the memory of my father Grégoire Krikorian (1934–2018)
This work was supported by a Chaire d’Excellence LABEX MME-DII, the project ANR BEKAM: ANR-15-CE40-0001 and an AAP project from CY Cergy Paris Université.
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Krikorian, R. On the divergence of Birkhoff Normal Forms. Publ.math.IHES 135, 1–181 (2022). https://doi.org/10.1007/s10240-022-00130-2
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DOI: https://doi.org/10.1007/s10240-022-00130-2