{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T22:28:59Z","timestamp":1773181739268,"version":"3.50.1"},"reference-count":11,"publisher":"American Mathematical Society (AMS)","issue":"12","license":[{"start":{"date-parts":[[2022,9,21]],"date-time":"2022-09-21T00:00:00Z","timestamp":1663718400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Proc. Amer. Math. Soc."],"abstract":"<p>\n                    We consider the almost reducibility property of a nonautonomous dynamics with discrete time defined by a sequence of matrices. This corresponds to the reduction of the original nonautonomous dynamics to an autonomous dynamics via a coordinate change that preserves the Lyapunov exponents. In particular, we give a characterization of the almost reducibility of a sequence to a diagonal matrix and we use this result to characterize the class of matrices to which a given sequence is almost reducible. We also consider continuous\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"1\">\n                        <mml:semantics>\n                          <mml:mn>1<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -parameter families of sequences of matrices and we show that the almost reducibility set of such a family is always an\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper F Subscript sigma delta\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>F<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>\n                                \u03c3\n                                \n                              <\/mml:mi>\n                              <mml:mi>\n                                \u03b4\n                                \n                              <\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">F_{\\sigma \\delta }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -set. In addition, we show that for any\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper F Subscript sigma delta\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>F<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>\n                                \u03c3\n                                \n                              <\/mml:mi>\n                              <mml:mi>\n                                \u03b4\n                                \n                              <\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">F_{\\sigma \\delta }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -set containing zero there exists a family with this set as its almost reducibility set.\n                  <\/p>","DOI":"10.1090\/proc\/15632","type":"journal-article","created":{"date-parts":[[2021,4,28]],"date-time":"2021-04-28T11:26:05Z","timestamp":1619609165000},"page":"5223-5236","source":"Crossref","is-referenced-by-count":1,"special_numbering":"750","title":["Almost reducibility for families of sequences of matrices"],"prefix":"10.1090","volume":"149","author":[{"given":"Luis","family":"Barreira","sequence":"first","affiliation":[]},{"given":"Claudia","family":"Valls","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2021,9,21]]},"reference":[{"issue":"8","key":"1","doi-asserted-by":"publisher","first-page":"1067","DOI":"10.1134\/S0012266109080011","article-title":"On the irregularity sets of families of linear differential systems","volume":"45","author":"Barabanov, E. A.","year":"2009","journal-title":"Differ. Uravn.","ISSN":"https:\/\/id.crossref.org\/issn\/0374-0641","issn-type":"print"},{"issue":"4","key":"2","doi-asserted-by":"publisher","first-page":"1750027","DOI":"10.1142\/S0219199717500274","article-title":"Transformations preserving the Lyapunov exponents","volume":"20","author":"Barreira, Luis","year":"2018","journal-title":"Commun. Contemp. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0219-1997","issn-type":"print"},{"issue":"4","key":"3","doi-asserted-by":"publisher","first-page":"1603","DOI":"10.1007\/s10884-019-09795-6","article-title":"Regularity and stability sets for families of sequences of matrices","volume":"32","author":"Barreira, Luis","year":"2020","journal-title":"J. Dynam. 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(2)","ISSN":"https:\/\/id.crossref.org\/issn\/0996-0481","issn-type":"print"},{"key":"8","doi-asserted-by":"publisher","first-page":"467","DOI":"10.2307\/1970195","article-title":"On almost periodic solutions of differential equations","volume":"69","author":"Lillo, James C.","year":"1959","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"key":"9","isbn-type":"print","volume-title":"The general problem of the stability of motion","author":"Lyapunov, A. M.","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/0748400621"},{"key":"10","doi-asserted-by":"publisher","first-page":"310","DOI":"10.1007\/BF01180637","article-title":"Continuous matrices and the stability of differential systems","volume":"62","author":"Markus, Lawrence","year":"1955","journal-title":"Math. 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