{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T14:10:35Z","timestamp":1772287835332,"version":"3.50.1"},"reference-count":20,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2008,8,1]],"date-time":"2008-08-01T00:00:00Z","timestamp":1217548800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ergod. Th. Dynam. Sys."],"published-print":{"date-parts":[[2008,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We consider the quadratic family of maps given by <jats:italic>f<\/jats:italic><jats:sub><jats:italic>a<\/jats:italic><\/jats:sub>(<jats:italic>x<\/jats:italic>)=1\u2212<jats:italic>ax<\/jats:italic><jats:sup>2<\/jats:sup> with <jats:italic>x<\/jats:italic>\u2208[\u22121,1], where <jats:italic>a<\/jats:italic> is a Benedicks\u2013Carleson parameter. For each of these chaotic dynamical systems we study the extreme value distribution of the stationary stochastic processes <jats:italic>X<\/jats:italic><jats:sub>0<\/jats:sub>,<jats:italic>X<\/jats:italic><jats:sub>1<\/jats:sub>,\u2026\u00a0, given by <jats:italic>X<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>=<jats:italic>f<\/jats:italic><jats:sub><jats:italic>a<\/jats:italic><\/jats:sub><jats:sup><jats:italic>n<\/jats:italic><\/jats:sup>, for every integer <jats:italic>n<\/jats:italic>\u22650, where each random variable <jats:italic>X<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> is distributed according to the unique absolutely continuous, invariant probability of <jats:italic>f<\/jats:italic><jats:sub><jats:italic>a<\/jats:italic><\/jats:sub>. Using techniques developed by Benedicks and Carleson, we show that the limiting distribution of <jats:italic>M<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>=max {<jats:italic>X<\/jats:italic><jats:sub>0<\/jats:sub>,\u2026,<jats:italic>X<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>\u22121<\/jats:sub>} is the same as that which would apply if the sequence <jats:italic>X<\/jats:italic><jats:sub>0<\/jats:sub>,<jats:italic>X<\/jats:italic><jats:sub>1<\/jats:sub>,\u2026 was independent and identically distributed. This result allows us to conclude that the asymptotic distribution of <jats:italic>M<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> is of type III (Weibull).<\/jats:p>","DOI":"10.1017\/s0143385707000624","type":"journal-article","created":{"date-parts":[[2008,3,12]],"date-time":"2008-03-12T06:24:56Z","timestamp":1205303096000},"page":"1117-1133","source":"Crossref","is-referenced-by-count":21,"title":["Extreme values for Benedicks\u2013Carleson quadratic maps"],"prefix":"10.1017","volume":"28","author":[{"given":"ANA CRISTINA","family":"MOREIRA FREITAS","sequence":"first","affiliation":[]},{"given":"JORGE MILHAZES","family":"FREITAS","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2008,8,1]]},"reference":[{"key":"S0143385707000624_ref11","doi-asserted-by":"publisher","DOI":"10.1007\/BF01941800"},{"key":"S0143385707000624_ref17","unstructured":"[17] Moreira F.\u00a0J. . Chaotic dynamics of quadratic maps, Informes de Matem\u00e1tica, IMPA, S\u00e9rie A, 092\/93, 1993, http:\/\/www.fc.up.pt\/cmup\/fsmoreir\/downloads\/BC.pdf."},{"key":"S0143385707000624_ref8","unstructured":"[8] Freitas J.\u00a0M. . Statistical stability for chaotic dynamical systems. PhD Thesis. Universidade do Porto, 2006, http:\/\/www.fc.up.pt\/pessoas\/jmfreita\/homeweb\/publications.htm."},{"key":"S0143385707000624_ref15","volume-title":"Extremes and Related Properties of Stationary Sequences and Processes","author":"Lindgren","year":"1983"},{"key":"S0143385707000624_ref1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02178361"},{"key":"S0143385707000624_ref5","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385700001802"},{"key":"S0143385707000624_ref18","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385700003199"},{"key":"S0143385707000624_ref6","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385701001201"},{"key":"S0143385707000624_ref4","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385700006556"},{"key":"S0143385707000624_ref7","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/18\/2\/019"},{"key":"S0143385707000624_ref2","doi-asserted-by":"publisher","DOI":"10.2307\/1971367"},{"key":"S0143385707000624_ref14","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511897214"},{"key":"S0143385707000624_ref10","doi-asserted-by":"publisher","DOI":"10.1016\/j.spl.2003.10.008"},{"key":"S0143385707000624_ref13","doi-asserted-by":"publisher","DOI":"10.1007\/BF02096623"},{"key":"S0143385707000624_ref12","doi-asserted-by":"publisher","DOI":"10.1007\/BF02773683"},{"key":"S0143385707000624_ref3","doi-asserted-by":"publisher","DOI":"10.2307\/2944326"},{"key":"S0143385707000624_ref16","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-78043-1"},{"key":"S0143385707000624_ref19","doi-asserted-by":"publisher","DOI":"10.1017\/S014338570000434X"},{"key":"S0143385707000624_ref9","unstructured":"[9] Freitas A.\u00a0C.\u00a0M. and Freitas J.\u00a0M. . On the link between dependence and independence in extreme value theory for dynamical systems. Statist. Probab. Lett. to appear."},{"key":"S0143385707000624_ref20","doi-asserted-by":"publisher","DOI":"10.1007\/BF02099211"}],"container-title":["Ergodic Theory and Dynamical Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0143385707000624","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,6]],"date-time":"2019-04-06T15:54:08Z","timestamp":1554566048000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0143385707000624\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,8]]},"references-count":20,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2008,8]]}},"alternative-id":["S0143385707000624"],"URL":"https:\/\/doi.org\/10.1017\/s0143385707000624","relation":{},"ISSN":["0143-3857","1469-4417"],"issn-type":[{"value":"0143-3857","type":"print"},{"value":"1469-4417","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,8]]}}}