Document Zbl 1286.37005

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Syst.</a> <a href="/?q=in%3A327833" title="Articles in this Issue">33, No. 6, 1786-1812 (2013)</a>. </div> <div class="abstract">In this paper, the authors systematically study dynamics of continuous maps on compact metrizable spaces containing a free or a disconnecting interval and compare the results with those arising from works on graphs or trees.<br class="zbmathjax-paragraph">There are three main results in this paper. The first two theorems deal with minimal systems and minimal sets, the third one is a dichotomy for transitive maps. Let $X$ be a compact metrizable space with a free interval $J$ and let $f : X \longrightarrow X$ be a continuous map. The results can be stated as follows:<br class="zbmathjax-paragraph">– If $f$ is minimal, then $X$ is a disjoint union of finitely many circles, $X=\oplus_{i=0}^{n-1}\mathbb{S}_i^1$, which are cyclically permuted by $f$ and, on each of them, $f^n$ is topologically conjugate to the same irrational rotation.<br class="zbmathjax-paragraph">– Assume that $M$ is a minimal set for $f$ which intersects $J$. Then exactly one of the following three statements holds. (1) $M$ is finite; (2) $M$ is a nowhere dense cantoroid; (3) $M$ is a disjoint union of finitely many circles.<br class="zbmathjax-paragraph">– Assume $f$ is transitive, then exactly one of the following two statements holds. (1) The map $f$ is relatively strongly mixing, non-invertible, has positive topological entropy and dense periodic points. (2) The space $X$ is a disjoint union of finitely many circles, $X=\oplus_{i=0}^{n-1}\mathbb{S}_i^1$, which are cyclically permuted by $f$ and, on each of them, $f^n$ is topologically conjugate to the same irrational rotation.<br class="zbmathjax-paragraph">The obtained results are potentially applicable in the study of dynamics on some classes of one-dimensional continua and some of them are even surprisingly nice.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A13996">Yujun Zhu (Shijiazhuang)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6251100">Cited in <strong>10</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><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><tr> <td> <a class="mono" href="/classification/?q=cc%3A37B40" title="MSC2020">37B40</a> </td> <td class="space"> Topological entropy </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Atopological+entropy">topological entropy</a>; <a href="/?q=ut%3Atransitive+system">transitive system</a>; <a href="/?q=ut%3Amixing+system">mixing system</a>; <a href="/?q=ut%3Adense+periodicity">dense periodicity</a>; <a href="/?q=ut%3Acontinuum">continuum</a>; <a href="/?q=ut%3Afree+interval">free interval</a>; <a href="/?q=ut%3Adisconnecting+interval">disconnecting interval</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 1286.37005" data-ciurl="/ci/06251100" data-biburl="/bibtex/06251100.bib" data-amsurl="/amsrefs/06251100.bib" data-xmlurl="/xml/06251100.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06251100.pdf" title="Zbl 1286.37005 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/10.1017/S0143385712000442" aria-label="DOI for “Minimality, transitivity, mixing and topological entropy on spaces with a free interval”" title="10.1017/S0143385712000442">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1112.1191"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Blokh, Differential-Difference Equations and Problems of Mathematical Physics pp 3– (1984)</td> </tr><tr> <td>[2]</td> <td class="space">DOI: 10.4064/cm93-1-9 · <a href="/1008.37006" class="nowrap">Zbl 1008.37006</a> · <a href="https://doi.org/10.4064/cm93-1-9" class="nowrap">doi:10.4064/cm93-1-9</a></td> </tr><tr> <td>[3]</td> <td class="space">DOI: 10.1142/S0218127403007576 · <a href="/1056.37050" class="nowrap">Zbl 1056.37050</a> · <a href="https://doi.org/10.1142/S0218127403007576" class="nowrap">doi:10.1142/S0218127403007576</a></td> </tr><tr> <td>[4]</td> <td class="space">Denjoy, J. 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