{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T14:46:07Z","timestamp":1772289967933,"version":"3.50.1"},"reference-count":39,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2012,8,6]],"date-time":"2012-08-06T00:00:00Z","timestamp":1344211200000},"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":[[2013,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We consider random perturbations of discrete-time dynamical systems. We give sufficient conditions for the stochastic stability of certain classes of maps, in a strong sense. This improves the main result in Alves and Ara\u00fajo [Random perturbations of non-uniformly expanding maps.<jats:italic>Ast\u00e9risque<\/jats:italic>\u00a0<jats:bold>286<\/jats:bold>(2003), 25\u201362], where the stochastic stability in the<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0143385712000077_inline1\"\/><jats:tex-math>$\\mathrm {weak}^*$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>topology was proved. Here, under slightly weaker assumptions on the random perturbations, we obtain a stronger version of stochastic stability: convergence of the density of the stationary measure to the density of the Sinai\u2013Ruelle\u2013Bowen (SRB) measure of the unperturbed system in the<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0143385712000077_inline2\"\/><jats:tex-math>$L^1$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-norm. As an application of our results, we obtain strong stochastic stability for two classes of non-uniformly expanding maps. The first one is an open class of local diffeomorphisms introduced in Alves, Bonatti and Viana [SRB measures for partially hyperbolic systems whose central direction is mostly expanding.<jats:italic>Invent. Math.<\/jats:italic>\u00a0<jats:bold>140<\/jats:bold>(2000), 351\u2013398] and the second one is the class of<jats:italic>Viana maps<\/jats:italic>.<\/jats:p>","DOI":"10.1017\/s0143385712000077","type":"journal-article","created":{"date-parts":[[2012,8,6]],"date-time":"2012-08-06T10:19:04Z","timestamp":1344248344000},"page":"647-692","source":"Crossref","is-referenced-by-count":16,"title":["Strong stochastic stability for non-uniformly expanding maps"],"prefix":"10.1017","volume":"33","author":[{"given":"JOS\u00c9 F.","family":"ALVES","sequence":"first","affiliation":[]},{"given":"HELDER","family":"VILARINHO","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2012,8,6]]},"reference":[{"key":"S0143385712000077_ref37","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385700003473"},{"key":"S0143385712000077_ref20","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385700006556"},{"key":"S0143385712000077_ref14","first-page":"179","article-title":"Random correlations for small perturbations of expanding maps","volume":"4","author":"Baladi","year":"1996","journal-title":"Random Comput. Dynam."},{"key":"S0143385712000077_ref34","doi-asserted-by":"publisher","DOI":"10.2307\/2373810"},{"key":"S0143385712000077_ref27","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-9175-3"},{"key":"S0143385712000077_ref25","doi-asserted-by":"publisher","DOI":"10.1007\/BF01941800"},{"key":"S0143385712000077_ref28","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4615-8181-9"},{"key":"S0143385712000077_ref21","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0081279"},{"key":"S0143385712000077_ref12","first-page":"77","article-title":"Almost sure rates of mixing for i.i.d. unimodal maps","volume":"35","author":"Baladi","year":"2002","journal-title":"Ann. Scient. \u00c9c. Norm. Sup., 4e s\u00e9rie"},{"key":"S0143385712000077_ref17","doi-asserted-by":"publisher","DOI":"10.2307\/1971367"},{"key":"S0143385712000077_ref39","doi-asserted-by":"publisher","DOI":"10.1023\/A:1019762724717"},{"key":"S0143385712000077_ref16","doi-asserted-by":"publisher","DOI":"10.1007\/BF02098487"},{"key":"S0143385712000077_ref36","doi-asserted-by":"publisher","DOI":"10.1007\/BF02699535"},{"key":"S0143385712000077_ref5","doi-asserted-by":"publisher","DOI":"10.1142\/S0219493707002049"},{"key":"S0143385712000077_ref18","doi-asserted-by":"publisher","DOI":"10.2307\/2944326"},{"key":"S0143385712000077_ref19","doi-asserted-by":"publisher","DOI":"10.1016\/j.anihpc.2004.10.013"},{"key":"S0143385712000077_ref11","doi-asserted-by":"publisher","DOI":"10.1007\/s002200050124"},{"key":"S0143385712000077_ref1","first-page":"1","article-title":"SRB measures for non-hyperbolic systems with multidimensional expansion","volume":"33","author":"Alves","year":"2000","journal-title":"Ann. Scient. \u00c9c. Norm. Sup., 4e s\u00e9rie"},{"key":"S0143385712000077_ref2","unstructured":"J.\u00a0F. Alves . Statistical analysis of non-uniformly expanding dynamical systems, 24th Braz. Math. Colloq., IMPA, Rio de Janeiro, 2003."},{"key":"S0143385712000077_ref38","doi-asserted-by":"publisher","DOI":"10.1007\/BF02808180"},{"key":"S0143385712000077_ref32","doi-asserted-by":"publisher","DOI":"10.2977\/prims\/1195182976"},{"key":"S0143385712000077_ref7","doi-asserted-by":"publisher","DOI":"10.1016\/j.anihpc.2004.12.002"},{"key":"S0143385712000077_ref4","first-page":"25","article-title":"Random perturbations of non-uniformly expanding maps","volume":"286","author":"Alves","year":"2003","journal-title":"Ast\u00e9risque"},{"key":"S0143385712000077_ref9","doi-asserted-by":"publisher","DOI":"10.1016\/S0294-1449(00)00112-8"},{"key":"S0143385712000077_ref13","doi-asserted-by":"crossref","unstructured":"V. Baladi , M. Benedicks and V. Maume-Deschamps . Correcting the proof of Theorem 3.2 in Almost sure rates of mixing for i.i.d. unimodal maps. Ann. E.N.S. (2002), arXiv:1008.5165 (2010).","DOI":"10.1016\/S0012-9593(03)00011-9"},{"key":"S0143385712000077_ref22","doi-asserted-by":"publisher","DOI":"10.1007\/BF01389848"},{"key":"S0143385712000077_ref15","first-page":"483","article-title":"Strong stochastic stability and rate of mixing for unimodal maps","volume":"29","author":"Baladi","year":"1996","journal-title":"Ann. Scient. \u00c9c. Norm. Sup., 4e s\u00e9rie"},{"key":"S0143385712000077_ref30","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0094308"},{"key":"S0143385712000077_ref26","doi-asserted-by":"publisher","DOI":"10.1007\/BF02792539"},{"key":"S0143385712000077_ref31","doi-asserted-by":"publisher","DOI":"10.1007\/s002200000220"},{"key":"S0143385712000077_ref35","doi-asserted-by":"publisher","DOI":"10.1070\/RM1972v027n04ABEH001383"},{"key":"S0143385712000077_ref3","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/17\/4\/004"},{"key":"S0143385712000077_ref6","doi-asserted-by":"publisher","DOI":"10.1007\/s002220000057"},{"key":"S0143385712000077_ref29","doi-asserted-by":"crossref","first-page":"83","DOI":"10.4064\/sm-33-1-83-92","article-title":"On invariant measures for expanding differentiable mappings","volume":"33","author":"Krzyzewski","year":"1969","journal-title":"Stud. Math."},{"key":"S0143385712000077_ref10","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-12878-7"},{"key":"S0143385712000077_ref33","first-page":"335","article-title":"A global view of dynamics and a conjecture on the denseness of finitude of attractors","volume":"261","author":"Palis","year":"2000","journal-title":"Ast\u00e9risque"},{"key":"S0143385712000077_ref8","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385702000019"},{"key":"S0143385712000077_ref23","doi-asserted-by":"publisher","DOI":"10.1007\/BF01198121"},{"key":"S0143385712000077_ref24","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385704000604"}],"container-title":["Ergodic Theory and Dynamical Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0143385712000077","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,25]],"date-time":"2023-06-25T05:28:11Z","timestamp":1687670891000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0143385712000077\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,8,6]]},"references-count":39,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2013,6]]}},"alternative-id":["S0143385712000077"],"URL":"https:\/\/doi.org\/10.1017\/s0143385712000077","relation":{},"ISSN":["0143-3857","1469-4417"],"issn-type":[{"value":"0143-3857","type":"print"},{"value":"1469-4417","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,8,6]]}}}