Document Zbl 1548.37088

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Math.</a> <a href="/?q=in%3A508146" title="Articles in this Issue">122, No. 2, 213-225 (2024)</a>. </div> <div class="abstract">Summary: It is known that a Diophantine quasi-periodic torus with frequency $\omega \in \Omega_{\tau}^d$ of a $C^l$ Hamiltonian is effectively stable for a time $T(r)$ that is polynomial on the inverse of the distance to the torus, that we denote by $r$, with exponent $1+(l-2)/(\tau+1)$. It is also known that a Diophantine quasi-periodic torus of a Gevrey Hamiltonian $H \in G^{\alpha, L}$ is effectively stable for an exponentially long time on the inverse of the distance to the torus with exponent $1/(\alpha (1+\tau))$. In this note, we see that following the methods in [the author and <span class="zbmathjax-textit">B. Fayad</span>, J. Eur. Math. Soc. (JEMS) 24, No. 12, 4363–4383 (2022; <a href="/1508.37072">Zbl 1508.37072</a>)] one can show the almost optimality of these exponents. We also show that, for a dense subset of non-resonant vectors, for quasi-periodic tori of finitely differentiable and Gevrey Hamiltonians, the naive lower bound $T(r) \geq Cr^{-1}$ is optimal in terms of the exponent.</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A37J40" title="MSC2020">37J40</a> </td> <td class="space"> Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37J25" title="MSC2020">37J25</a> </td> <td class="space"> Stability problems for finite-dimensional Hamiltonian and Lagrangian systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A70K43" title="MSC2020">70K43</a> </td> <td class="space"> Quasi-periodic motions and invariant tori for nonlinear problems in mechanics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A70H08" title="MSC2020">70H08</a> </td> <td class="space"> Nearly integrable Hamiltonian systems, KAM theory </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AHamiltonian+systems">Hamiltonian systems</a>; <a href="/?q=ut%3Aquasi-periodic+tori">quasi-periodic tori</a>; <a href="/?q=ut%3Aeffective+stability">effective stability</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1508.37072">Zbl 1508.37072</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1548.37088" data-ciurl="/ci/07801706" data-biburl="/bibtex/07801706.bib" data-amsurl="/amsrefs/07801706.bib" data-xmlurl="/xml/07801706.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07801706.pdf" title="Zbl 1548.37088 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s00013-023-01928-0" aria-label="DOI for “On the optimal effective stability bounds for quasi-periodic tori of finitely differentiable and Gevrey Hamiltonians”" title="10.1007/s00013-023-01928-0">DOI</a> </div> <div class="sfx" style="float: right;"> <a href="https://creativecommons.org/licenses/by/4.0/" target="_blank" title="Open Access License" class="cc-license-link no-new-tab-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/cc.svg" alt="Creative Commons CC license icon" class="cc-license-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/by.svg" alt="Creative Commons BY license icon" class="cc-license-icon"> </a> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Anosov, DV; Katok, AB, New examples in smooth ergodic theory, Ergodic diffeomorphisms. 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