{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T14:30:39Z","timestamp":1773239439838,"version":"3.50.1"},"reference-count":9,"publisher":"World Scientific Pub Co Pte Lt","issue":"11","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2012,11]]},"abstract":"<jats:p> Metastable dynamical systems were recently studied [Gonz\u00e1lez-Tokman et al., 2011] in the framework of one-dimensional piecewise expanding maps on two disjoint invariant sets, each possessing its own ergodic absolutely continuous invariant measure (acim). Under small deterministic perturbations, holes between the two disjoint systems are created, and the two ergodic systems merge into one. The long term dynamics of the newly formed metastable system is defined by the unique acim on the combined ergodic sets. The main result of [Gonz\u00e1lez-Tokman et al., 2011] proves that this combined acim can be approximated by a convex combination of the disjoint acims with weights depending on the ratio of the respective measures of the holes. In this note we present an entirely different approach to metastable systems. We consider two piecewise expanding maps: one is the original map, \u03c4<jats:sub>1<\/jats:sub>, defined on two disjoint invariant sets of \u211d<jats:sup>N<\/jats:sup> and the other is a deterministically perturbed version of \u03c4<jats:sub>1<\/jats:sub>, \u03c4<jats:sub>2<\/jats:sub>, which allows passage between the two disjoint invariant sets of \u03c4<jats:sub>1<\/jats:sub>. We model this system by a position dependent random map based on \u03c4<jats:sub>1<\/jats:sub> and \u03c4<jats:sub>2<\/jats:sub>, to which we associate position dependent probabilities that reflect the switching between the maps. A typical orbit spends a long time in one of the ergodic sets but eventually switches to the other. Such behavior can be attributed to physical holes as between adjoining billiard tables or more abstract situations where balls can \"leap\" from one table to the other. Using results for random maps, a result similar to the one-dimensional main result of [Gonz\u00e1lez-Tokman et al., 2011] is proved in N dimensions. We also consider holes in more than two invariant sets. A number of examples are presented. <\/jats:p>","DOI":"10.1142\/s0218127412502793","type":"journal-article","created":{"date-parts":[[2012,12,13]],"date-time":"2012-12-13T09:14:17Z","timestamp":1355390057000},"page":"1250279","source":"Crossref","is-referenced-by-count":3,"title":["METASTABLE SYSTEMS AS RANDOM MAPS"],"prefix":"10.1142","volume":"22","author":[{"given":"PAWE\u0141","family":"G\u00d3RA","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"ABRAHAM","family":"BOYARSKY","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"PEYMAN","family":"ESLAMI","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2012,12,13]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.4064\/sm166-3-5"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-2024-4"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(01)00320-2"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1142\/S0219493712003547"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-9486-0"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385710000337"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1007\/BF02764946"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-247X(02)00658-3"},{"key":"rf9","first-page":"141","volume":"28","author":"Keller G.","journal-title":"Ann. Scuola Norm. Sup. Pisa Cl. Sci."}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127412502793","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T18:50:21Z","timestamp":1565117421000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127412502793"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,11]]},"references-count":9,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2012,12,13]]},"published-print":{"date-parts":[[2012,11]]}},"alternative-id":["10.1142\/S0218127412502793"],"URL":"https:\/\/doi.org\/10.1142\/s0218127412502793","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,11]]}}}