article published yes Abed Bounemoura author Vadim Kaloshin author FE553552-CDE8-11E9-B324-C0EBE56974250000-0002-6051-2628 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. American Mathematical Society2015 eng 0002-9939 1088-682610.1090/proc/12796 14441553-1560 yes A. Bounemoura, V. Kaloshin, Proceedings of the American Mathematical Society 144 (2015) 1553–1560. Bounemoura, Abed, and Vadim Kaloshin. “A Note on Micro-Instability for Hamiltonian Systems Close to Integrable.” <i>Proceedings of the American Mathematical Society</i>. American Mathematical Society, 2015. <a href="https://doi.org/10.1090/proc/12796">https://doi.org/10.1090/proc/12796</a>. Bounemoura A, Kaloshin V. A note on micro-instability for Hamiltonian systems close to integrable. <i>Proceedings of the American Mathematical Society</i>. 2015;144(4):1553-1560. doi:<a href="https://doi.org/10.1090/proc/12796">10.1090/proc/12796</a> A. Bounemoura and V. Kaloshin, “A note on micro-instability for Hamiltonian systems close to integrable,” <i>Proceedings of the American Mathematical Society</i>, vol. 144, no. 4. American Mathematical Society, pp. 1553–1560, 2015. Bounemoura, A., &#38; Kaloshin, V. (2015). A note on micro-instability for Hamiltonian systems close to integrable. <i>Proceedings of the American Mathematical Society</i>. American Mathematical Society. <a href="https://doi.org/10.1090/proc/12796">https://doi.org/10.1090/proc/12796</a> Bounemoura A, Kaloshin V. 2015. A note on micro-instability for Hamiltonian systems close to integrable. Proceedings of the American Mathematical Society. 144(4), 1553–1560. Bounemoura, Abed, and Vadim Kaloshin. “A Note on Micro-Instability for Hamiltonian Systems Close to Integrable.” <i>Proceedings of the American Mathematical Society</i>, vol. 144, no. 4, American Mathematical Society, 2015, pp. 1553–60, doi:<a href="https://doi.org/10.1090/proc/12796">10.1090/proc/12796</a>. 84952020-09-18T10:46:14Z2021-01-12T08:19:40Z