Document Zbl 1144.37002

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Dynamics of systems with nonzero Lyapunov exponents.</strong> <i>(English)</i> <a class="label nowrap" href="/1144.37002">Zbl 1144.37002</a> </h2> <div class="source"> <a href="/serials/3879" title="Series Profile">Encyclopedia of Mathematics and its Applications</a> 115. Cambridge: Cambridge University Press (ISBN 978-0-521-83258-8/hbk). xiv, 513 p. (2007). </div> <div class="abstract">Nonuniform hyperbolicity theory is an important part of the general theory of dynamical systems. Its core is the study of dynamical systems with nonzero Lyapunov exponents both conservative and dissipative, in addition to cocycles and group actions. The results of this theory are widely used in geometry (e.g., geodesic flows and Teichm眉ller flows), in rigidity theory, in the study of some partial differential equations (e.g., the Schr枚dinger equation), and in the theory of billiards, as well as in applications to physics, biology, engineering, and other fields. Among other things, the nonuniform hyperbolicity theory provides a rigorous mathematical foundation for the phenomenon known as deterministic chaos – the appearance of chaotic motions in pure deterministic dynamical systems. A sufficiently complete description of topological and ergodic properties of systems exhibiting deterministic chaos can be deduced from relatively weak requirements on their local behavior known as nonuniform hyperbolicity conditions. <br class="zbmathjax-paragraph"> The book presents a self-contained and sufficiently complete description of the modern nonuniform hyperbolicity theory and comprehensive account of modern smooth ergodic theory. It contains the core results of the theories as well as an account of their recent developments. Despite a tremendous amount of research on the subject, there have been only few attempts to summarize and unify the results of the theories in a single manuscript or a survey; this book covers this gap. <br class="zbmathjax-paragraph"> The book consists of four parts divided into 15 chapters. In more detail, the material of the book can be viewed from its contents. <dl class="zbmathjax-itemize"><dt class="zbmathjax-item1">●</dt><dd class="zbmathjax-item1">{}Part I. Linear theory </dd><dt class="zbmathjax-item2">1.</dt><dd class="zbmathjax-item2">The concept of nonuniform hyperbolicity. </dd><dt class="zbmathjax-item2">2.</dt><dd class="zbmathjax-item2">Lyapunov exponents for linear extensions. </dd><dt class="zbmathjax-item2">3.</dt><dd class="zbmathjax-item2">Regularity of cocycles. </dd><dt class="zbmathjax-item2">4.</dt><dd class="zbmathjax-item2">Methods for estimating exponents. </dd><dt class="zbmathjax-item2">5.</dt><dd class="zbmathjax-item2">The derivative cocycles. </dd><dt class="zbmathjax-item1">●</dt><dd class="zbmathjax-item1">{}Part II. Examples and foundation of the nonlinear theory </dd><dt class="zbmathjax-item2">6.</dt><dd class="zbmathjax-item2">Examples of systems with hyperbolic behavior. </dd><dt class="zbmathjax-item2">7.</dt><dd class="zbmathjax-item2">Stable manifold theory. </dd><dt class="zbmathjax-item2">8.</dt><dd class="zbmathjax-item2">Basic properties of stable and unstable manifolds. </dd><dt class="zbmathjax-item1">●</dt><dd class="zbmathjax-item1">{}Part III. Ergodic theory of smooth and SRB measures </dd><dt class="zbmathjax-item2">9.</dt><dd class="zbmathjax-item2">Smooth measures. </dd><dt class="zbmathjax-item2">10.</dt><dd class="zbmathjax-item2">Measure-theoretic entropy and Lyapunov exponents. </dd><dt class="zbmathjax-item2">11.</dt><dd class="zbmathjax-item2">Stable ergodicity and Lyapunov exponents. More examples of systems with nonzero exponents. </dd><dt class="zbmathjax-item2">12.</dt><dd class="zbmathjax-item2">Geodesic flows. </dd><dt class="zbmathjax-item2">13.</dt><dd class="zbmathjax-item2">SRB measures. </dd><dt class="zbmathjax-item1">●</dt><dd class="zbmathjax-item1">{}Part IV General hyperbolic measures </dd><dt class="zbmathjax-item2">14.</dt><dd class="zbmathjax-item2">Hyperbolic measures: entropy and dimension. </dd><dt class="zbmathjax-item2">15.</dt><dd class="zbmathjax-item2">Hyperbolic measures: topological properties. </dd></dl> <br class="zbmathjax-paragraph">All the principle results of the nonuniform hyperbolicity theory are presented in the book with complete proofs, although some other results are included without proofs. Most chapters of the book end with notes in which the reader can find some remarks of historical and bibliographical nature, comments on some related results, and references for further reading. <br class="zbmathjax-paragraph"> Beautifully written, the book will be useful as a reference and as a supplement to an advanced course on dynamical systems.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A2863">Georgy Osipenko (St. Peterburg)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5218477">Cited in <strong>279</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A37-02" title="MSC2020">37-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37D25" title="MSC2020">37D25</a> </td> <td class="space"> Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37A30" title="MSC2020">37A30</a> </td> <td class="space"> Ergodic theorems, spectral theory, Markov operators </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37D10" title="MSC2020">37D10</a> </td> <td class="space"> Invariant manifold theory for dynamical systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37A20" title="MSC2020">37A20</a> </td> <td class="space"> Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Astable+manifold+theory">stable manifold theory</a>; <a href="/?q=ut%3Aergodic+theory">ergodic theory</a>; <a href="/?q=ut%3ASRB+measures">SRB measures</a>; <a href="/?q=ut%3Ameasure-theoretic+entropy">measure-theoretic entropy</a>; <a href="/?q=ut%3Ageodesic+flows">geodesic flows</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 1144.37002" data-ciurl="/ci/05218477" data-biburl="/bibtex/05218477.bib" data-amsurl="/amsrefs/05218477.bib" data-xmlurl="/xml/05218477.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05218477.pdf" title="Zbl 1144.37002 as PDF">Review PDF</a> </div> <div class="fulltexts"> </div> <div class="sfx" style="float: right;"> </div> </div> </article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; 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