Document Zbl 1246.47004

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Berlin: Springer (ISBN 978-1-4471-2169-5/pbk; 978-1-4471-2170-1/ebook). xii, 386 p. (2011). </div> <div class="abstract">The book under review aims at presenting in an accessible and self-contained way the theory of hypercyclicity and, more generally, the theory of chaos for linear dynamical systems. Linear dynamical systems form what may at first sight be perceived as a very particular subclass of dynamical systems: these are systems of the form $(X,T)$, where $X$ is a separable Banach or Fréchet space, and $T$ is a continuous linear operator on $X$. The richness of this class of systems appears when $X$ is infinite-dimensional and, of course, in this case the underlying space is not locally compact. On the other hand, the linearity of $T$ is a strong assumption which gives rise to all kinds of unexpected phenomena.<br class="zbmathjax-paragraph">The study of these linear dynamical systems really started in the 1970’s with Kitai’s thesis, and it has developed since then in many directions. One of the main reasons why linear dynamical systems are the object of such an interest is that they exhibit in the linear framework features which are intuitively thought of as being nonlinear (the prototype of this being chaotic behavior). One can construct, more or less easily, examples of topologically transitive, chaotic, topologically mixing, ergodic, weakly mixing, strongly mixing, weakly mixing but not strongly mixing, minimal, etc., linear dynamical systems, and a wide range of properties appear as properties of such systems. The linearity of the operator $T$, which may look as a rigidity property which would prevent the system from exhibiting any kind of chaotic behavior, is counterbalanced by the fact that the space is infinite-dimensional; this interplay lies at the core of the theory.<br class="zbmathjax-paragraph">The book under review does an excellent job of presenting this theory, assuming only basic knowledge of functional and complex analysis. It is accessible to advanced undergraduate or beginning graduate students, as well as to nonspecialists who wish to learn about linear dynamical systems. The book covers in great detail what may now be thought of as the basis of the theory, insisting on examples. It deliberately does not cover some more advanced material (such as the ergodic theory of linear dynamical systems), and is in this sense complementary to the book “Linear Dynamical Systems” by <span class="zbmathjax-textit">F. Bayart</span> and <span class="zbmathjax-textit">É. Matheron</span> [Dynamics of linear operators. Cambridge Tracts in Mathematics 179, Cambridge University Press (2009; <a href="/1187.47001">Zbl 1187.47001</a>)]. Let us now proceed to a more detailed description of the contents of the book.<br class="zbmathjax-paragraph">The first chapter presents some elementary material on (nonlinear) topological dynamics, the crucial notions being those of topological transitivity, chaos, and weak and strong mixing. These notions are at the core of the second chapter, but this time in the linear setting: since the underlying space $X$ of a linear dynamical system $(X,T)$ is always assumed to be complete, $T$ is topologically transitive if and only if it admits a vector $x\in T$ whose orbit $\text{Orb}(x,T)=\{T^{n}x\mid\;n\geq 0\}$ is dense in $X$. Such vectors are called <span class="zbmathjax-textit">hypercyclic vectors</span> for $T$, and the operator $T$ itself is said to be <span class="zbmathjax-textit">hypercyclic</span> if it admits a (and hence a residual set of) hypercyclic vector(s). Devaney’s chaos for linear systems can be characterized in this setting as follows: $T$ is <span class="zbmathjax-textit">chaotic</span> if and only if it is hypercyclic and has a dense set of periodic vectors. The first examples of hypercyclic and chaotic operators appear in the second chapter: <dl class="zbmathjax-itemize"><dt class="zbmathjax-item1">–</dt><dd class="zbmathjax-item1">the operator $D:f\longmapsto f'$ of differentiation on the space $H(\mathbb{C})$ of entire functions on $\mathbb{C}$ (MacLane’s operator); </dd><dt class="zbmathjax-item1">–</dt><dd class="zbmathjax-item1">the operators $T_{a}:f\longmapsto f(.+a)$, $a\neq 0$, of translation on $H(\mathbb{C})$ (Birkhoff’s operators); </dd><dt class="zbmathjax-item1">–</dt><dd class="zbmathjax-item1">multiples of $\lambda B$, $|\lambda |>1$, of the backward shift $B$ on $\ell_{2}(\mathbb{N})$ (Rolewicz’s operators). </dd></dl> Chapter 3 is central in the book. It presents the so-called Hypercyclicity Criterion. This is a very efficient criterion for proving that an operator is indeed hypercyclic, and it uses linearity in a crucial way. Many equivalent formulations of this fundamental criterion are given, as well as one of its most important applications, the Godefroy-Shapiro Criterion, which gives conditions bearing on the eigenvectors of the operator under which it is hypercyclic or chaotic.<br class="zbmathjax-paragraph">Now that the stage is set, Chapter 4 presents a huge variety of examples of hypercyclic and chaotic operators (uni- or bilateral weighted shifts, differential operators, composition operators on spaces of holomorphic functions or on the Hardy space $H^{2}(\mathbb{D})$, adjoints of multipliers on $H^{2}(\mathbb{D})$, etc.), while Chapter 5 exhibits necessary conditions for an operator to be hypercyclic. For instance, any connected component of the spectrum of a hypercyclic operator must intersect the unit circle.<br class="zbmathjax-paragraph">Chapter 6, which ends the first part of the book, focuses on four important results whose proofs all rely on a connectedness argument. These four results exhibit a kind of rigidity in hypercyclic systems. There is at first Ansari’s result that all powers $T^{p}$, $p\geq 2$, of a hypercyclic operator $T$ are hypercyclic, then the Bourdon-Feldman theorem that an orbit $\text{Orb}(x,T)$ of an operator $T$ which is somewhere dense in $X$ is automatically dense in $X$, and finally two results whose proofs fit into a common framework, that of semigroup actions. First: if $T$ is a hypercyclic operator, any rotation $\lambda T$ of $T$, $|\lambda |=1$, is hypercyclic and the set of its hypercyclic vectors coincides with that of $T$ (this is a result of León-Müller). Second: Suppose that $(T_{t})_{t\geq 0}$ is a $C_{0}$-semigroup on a Banach space $X$, and that $x\in X$ is a vector which is hypercyclic for $T_{t}$ (i.e., such that $\{T_{t}x\mid t\geq 0\}$ is dense in $X$), then $x$ is a hypercyclic vector for each individual operator $T_{t}$, $t>0$ (this is a result of Conejero-Müller-Peris).<br class="zbmathjax-paragraph"> The second part of the book covers selected topics, and may be read in any order. Chapter 7 focuses on hypercyclic and chaotic semigroups of operators. Chapter 8 presents existence results, in particular the striking Ansari-Bernal result that any separable infinite-dimensional Banach space $X$ supports a hypercyclic operator. Chapter 9 is devoted to a strengthened form of hypercyclicity, which is called <span class="zbmathjax-textit">frequent hypercyclicity</span> and has links with the ergodic theory of linear dynamical systems. A vector $x\in X$ is a <span class="zbmathjax-textit">frequently hypercyclic vector</span> for $T$ if for any nonempty open subset $U$ of $X$ the set ${\{n\geq 0\mid T^{n}x\in U\}}$ has positive lower density. Chapter 10 investigates conditions under which an operator admits an infinite-dimensional closed subspace which consists entirely, except for $0$, of hypercyclic vectors. Conditions under which an uncountable family of hypercyclic operators admits a common hypercyclic vector are studied in Chapter 11, and the last chapter investigates which kind of results in linear dynamics extend to the setting of topological vector spaces (where the Baire Category Theorem is no longer available).<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A11047">Sophie Grivaux (Villeneuve d’Ascq)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A5943676">Cited in <strong>2</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A5943676">Cited in <strong>587</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A47A16" title="MSC2020">47A16</a> </td> <td class="space"> Cyclic vectors, hypercyclic and chaotic operators </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A47D06" title="MSC2020">47D06</a> </td> <td class="space"> One-parameter semigroups and linear evolution equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37B05" title="MSC2020">37B05</a> </td> <td class="space"> Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Alinear+dynamical+systems">linear dynamical systems</a>; <a href="/?q=ut%3Ahypercyclic+and+chaotic+operators">hypercyclic and chaotic operators</a>; <a href="/?q=ut%3Adynamics+of+semigroups">dynamics of semigroups</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1187.47001">Zbl 1187.47001</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 1246.47004" data-ciurl="/ci/05943676" data-biburl="/bibtex/05943676.bib" data-amsurl="/amsrefs/05943676.bib" data-xmlurl="/xml/05943676.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05943676.pdf" title="Zbl 1246.47004 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/10.1007/978-1-4471-2170-1" aria-label="DOI for “Linear chaos”" title="10.1007/978-1-4471-2170-1">DOI</a> </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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