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A note on micro-instability for Hamiltonian systems close to integrable


Authors: Abed Bounemoura and Vadim Kaloshin
Journal: Proc. Amer. Math. Soc. 144 (2016), 1553-1560
MSC (2010): Primary 37J25, 37J40
DOI: https://doi.org/10.1090/proc/12796
Published electronically: December 21, 2015
MathSciNet review: 3451232
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Abstract: In this note, we consider the dynamics associated to a perturbation of an integrable Hamiltonian system in action-angle coordinates in any number of degrees of freedom and we prove the following result of ``micro-diffusion'': under generic assumptions on $ h$ and $ f$, there exists an orbit of the system for which the drift of its action variables is at least of order $ \sqrt {\varepsilon }$, after a time of order $ \sqrt {\varepsilon }^{-1}$. The assumptions, which are essentially minimal, are that there exists a resonant point for $ h$ and that the corresponding averaged perturbation is non-constant. The conclusions, although very weak when compared to usual instability phenomena, are also essentially optimal within this setting.


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Additional Information

Abed Bounemoura
Affiliation: CNRS-IMPA UMI, Rio de Janeiro AC 22460-320, Brazil
Email: abedbou@gmail.com

Vadim Kaloshin
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: vadim.kaloshin@gmail.com

DOI: https://doi.org/10.1090/proc/12796
Received by editor(s): December 19, 2014
Received by editor(s) in revised form: March 30, 2015
Published electronically: December 21, 2015
Communicated by: Yingfei Yi
Article copyright: © Copyright 2015 American Mathematical Society