{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,5]],"date-time":"2026-08-05T22:41:26Z","timestamp":1785969686375,"version":"3.56.0"},"reference-count":18,"publisher":"World Scientific Pub Co Pte Lt","issue":"04","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2009,4]]},"abstract":"<jats:p> Let X be a dendrite and f : X \u2192 X be a continuous map. Denote by R(f) and P(f) the sets of recurrent points and periodic points of f respectively. In this paper we show that, if the cardinal number Card ( End (X)) of the set of endpoints of X is less than the cardinal number c of the continuum, then [Formula: see text]. From this we derive that, if Card ( End (X)) &lt; c, then f is chaotic in the sense of Devaney if and only if f is transitive. <\/jats:p>","DOI":"10.1142\/s021812740902372x","type":"journal-article","created":{"date-parts":[[2009,8,17]],"date-time":"2009-08-17T05:02:26Z","timestamp":1250485346000},"page":"1391-1396","source":"Crossref","is-referenced-by-count":46,"title":["$\\overline{R} = \\overline{P}$ FOR MAPS OF DENDRITES X WITH <font>Card<\/font>(<font>End<\/font>(X)) &lt; c"],"prefix":"10.1142","volume":"19","author":[{"given":"JIEHUA","family":"MAI","sequence":"first","affiliation":[{"name":"Institute of Mathematics, Shantou University, Shantou 515063, P. R. China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"ENHUI","family":"SHI","sequence":"additional","affiliation":[{"name":"Mathematics and Sciences College, Suzhou University, Suzhou 215006, P. R. China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","first-page":"753","volume":"33","author":"Acosta G.","journal-title":"Houston J. Math."},{"key":"rf2","first-page":"1551","volume":"153","author":"Alsed\u00e0 Ll.","journal-title":"Trans. Amer. Math. Soc."},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127495000971"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1016\/j.topol.2007.04.001"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127403007576"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.2307\/2324899"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4422-6"},{"key":"rf8","first-page":"316","volume":"79","author":"Coven E. M.","journal-title":"Proc. Amer. Math. Soc."},{"key":"rf9","volume-title":"An Introduction to Chaotic Dynamical Systems","author":"Devaney R. L.","year":"1989"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1070\/SM2001v192n06ABEH000570"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1080\/1023619021000047806"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1017\/S0004972700014283"},{"key":"rf13","first-page":"3","volume":"334","author":"Kolyada S.","journal-title":"Grazer Math. Ber., Bericht Nr."},{"key":"rf14","doi-asserted-by":"publisher","DOI":"10.1201\/9781315274089"},{"key":"rf15","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61717-1"},{"key":"rf16","doi-asserted-by":"publisher","DOI":"10.2307\/2975629"},{"key":"rf17","first-page":"294","volume":"39","author":"Xiong J. C.","journal-title":"Acta Math. Sinica, Chin. Ser."},{"key":"rf18","doi-asserted-by":"publisher","DOI":"10.1017\/S0004972700015768"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S021812740902372X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T20:02:45Z","timestamp":1565121765000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S021812740902372X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,4]]},"references-count":18,"journal-issue":{"issue":"04","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2009,4]]}},"alternative-id":["10.1142\/S021812740902372X"],"URL":"https:\/\/doi.org\/10.1142\/s021812740902372x","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,4]]}}}