{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T17:50:42Z","timestamp":1775065842927,"version":"3.50.1"},"reference-count":14,"publisher":"IOP Publishing","issue":"3","license":[{"start":{"date-parts":[[2013,2,13]],"date-time":"2013-02-13T00:00:00Z","timestamp":1360713600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/publishingsupport.iopscience.iop.org\/iop-standard\/v1"},{"start":{"date-parts":[[2013,2,13]],"date-time":"2013-02-13T00:00:00Z","timestamp":1360713600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/iopscience.iop.org\/info\/page\/text-and-data-mining"}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"short-container-title":["Nonlinearity"],"published-print":{"date-parts":[[2013,3,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We show that if the Lyapunov exponents associated with a sequence of matrices\n                    <jats:inline-formula>\n                      <jats:tex-math>\n                        \n                      <\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    are limits, then this asymptotic behaviour is mimicked by the sequences\n                    <jats:italic>x<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>m<\/jats:italic>\n                      +1\n                    <\/jats:sub>\n                    \u00a0=\u00a0\n                    <jats:italic>A<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>m<\/jats:italic>\n                    <\/jats:sub>\n                    <jats:italic>x<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>m<\/jats:italic>\n                    <\/jats:sub>\n                    \u00a0+\u00a0\n                    <jats:italic>f<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>m<\/jats:italic>\n                    <\/jats:sub>\n                    (\n                    <jats:italic>x<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>m<\/jats:italic>\n                    <\/jats:sub>\n                    ) for any sufficiently small continuous perturbations\n                    <jats:italic>f<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>m<\/jats:italic>\n                    <\/jats:sub>\n                    . Moreover, no new values of the Lyapunov exponents are introduced. Our approach is based on Lyapunov's theory of regularity, including its generalization to infinite-dimensional spaces.\n                  <\/jats:p>","DOI":"10.1088\/0951-7715\/26\/3\/855","type":"journal-article","created":{"date-parts":[[2013,2,13]],"date-time":"2013-02-13T07:23:53Z","timestamp":1360740233000},"page":"855-870","update-policy":"https:\/\/doi.org\/10.1088\/crossmark-policy","source":"Crossref","is-referenced-by-count":5,"title":["RETRACTED: A Perron-type theorem for nonautonomous difference equations"],"prefix":"10.1088","volume":"26","author":[{"given":"Luis","family":"Barreira","sequence":"first","affiliation":[]},{"given":"Claudia","family":"Valls","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2013,2,13]]},"reference":[{"key":"non443817bib01","author":"Barreira","year":"2002","type":"book"},{"key":"non443817bib02","doi-asserted-by":"publisher","author":"Barreira","year":"2007","DOI":"10.1017\/CBO9781107326026","type":"book"},{"key":"non443817bib03","doi-asserted-by":"publisher","first-page":"204","DOI":"10.1016\/j.jde.2005.05.008","type":"journal-article","article-title":"Stability of nonautonomous differential equations in Hilbert spaces","volume":"217","author":"Barreira","year":"2005","journal-title":"J. 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