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Appl."],"published-print":{"date-parts":[[2024,5]]},"abstract":"Abstract<\/jats:title>In the present paper we prove that densely, with respect to an $$L^p$$<\/jats:tex-math>\n \n L<\/mml:mi>\n p<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-like topology, the Lyapunov exponents associated to linear continuous-time cocycles $$\\Phi :\\mathbb {R}\\times M\\rightarrow {{\\,\\textrm{GL}\\,}}(2,\\mathbb {R})$$<\/jats:tex-math>\n \n \u03a6<\/mml:mi>\n :<\/mml:mo>\n R<\/mml:mi>\n \u00d7<\/mml:mo>\n M<\/mml:mi>\n \u2192<\/mml:mo>\n \n \n GL<\/mml:mtext>\n \n <\/mml:mrow>\n (<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n R<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> induced by second order linear homogeneous differential equations $$\\ddot{x}+\\alpha (\\varphi ^t(\\omega ))\\dot{x}+\\beta (\\varphi ^t(\\omega ))x=0$$<\/jats:tex-math>\n \n \n x<\/mml:mi>\n \u00a8<\/mml:mo>\n <\/mml:mover>\n +<\/mml:mo>\n \u03b1<\/mml:mi>\n \n (<\/mml:mo>\n \n \u03c6<\/mml:mi>\n t<\/mml:mi>\n <\/mml:msup>\n \n (<\/mml:mo>\n \u03c9<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n \n x<\/mml:mi>\n \u02d9<\/mml:mo>\n <\/mml:mover>\n +<\/mml:mo>\n \u03b2<\/mml:mi>\n \n (<\/mml:mo>\n \n \u03c6<\/mml:mi>\n t<\/mml:mi>\n <\/mml:msup>\n \n (<\/mml:mo>\n \u03c9<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n x<\/mml:mi>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are almost everywhere distinct. The coefficients $$\\alpha ,\\beta $$<\/jats:tex-math>\n \n \u03b1<\/mml:mi>\n ,<\/mml:mo>\n \u03b2<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> evolve along the $$\\varphi ^t$$<\/jats:tex-math>\n \n \u03c6<\/mml:mi>\n t<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-orbit for $$\\omega \\in M$$<\/jats:tex-math>\n \n \u03c9<\/mml:mi>\n \u2208<\/mml:mo>\n M<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$\\varphi ^t: M\\rightarrow M$$<\/jats:tex-math>\n \n \n \u03c6<\/mml:mi>\n t<\/mml:mi>\n <\/mml:msup>\n :<\/mml:mo>\n M<\/mml:mi>\n \u2192<\/mml:mo>\n M<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is an ergodic flow defined on a probability space. We also obtain the corresponding version for the frictionless equation $$\\ddot{x}+\\beta (\\varphi ^t(\\omega ))x=0$$<\/jats:tex-math>\n \n \n x<\/mml:mi>\n \u00a8<\/mml:mo>\n <\/mml:mover>\n +<\/mml:mo>\n \u03b2<\/mml:mi>\n \n (<\/mml:mo>\n \n \u03c6<\/mml:mi>\n t<\/mml:mi>\n <\/mml:msup>\n \n (<\/mml:mo>\n \u03c9<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n x<\/mml:mi>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and for a Schr\u00f6dinger equation $$\\ddot{x}+(E-Q(\\varphi ^t(\\omega )))x=0$$<\/jats:tex-math>\n \n \n x<\/mml:mi>\n \u00a8<\/mml:mo>\n <\/mml:mover>\n +<\/mml:mo>\n \n (<\/mml:mo>\n E<\/mml:mi>\n -<\/mml:mo>\n Q<\/mml:mi>\n \n (<\/mml:mo>\n \n \u03c6<\/mml:mi>\n t<\/mml:mi>\n <\/mml:msup>\n \n (<\/mml:mo>\n \u03c9<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n x<\/mml:mi>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, inducing a cocycle $$\\Phi :\\mathbb {R}\\times M\\rightarrow {{\\,\\textrm{SL}\\,}}(2,\\mathbb {R})$$<\/jats:tex-math>\n \n \u03a6<\/mml:mi>\n :<\/mml:mo>\n R<\/mml:mi>\n \u00d7<\/mml:mo>\n M<\/mml:mi>\n \u2192<\/mml:mo>\n \n \n SL<\/mml:mtext>\n \n <\/mml:mrow>\n (<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n R<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.\n<\/jats:p>","DOI":"10.1007\/s00030-024-00931-w","type":"journal-article","created":{"date-parts":[[2024,3,24]],"date-time":"2024-03-24T12:01:26Z","timestamp":1711281686000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Simple Lyapunov spectrum for linear homogeneous differential equations with $$L^p$$ parameters"],"prefix":"10.1007","volume":"31","author":[{"given":"Dinis","family":"Amaro","sequence":"first","affiliation":[]},{"given":"M\u00e1rio","family":"Bessa","sequence":"additional","affiliation":[]},{"given":"Helder","family":"Vilarinho","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,3,24]]},"reference":[{"issue":"3","key":"931_CR1","doi-asserted-by":"publisher","first-page":"723","DOI":"10.2307\/1969259","volume":"42","author":"W Ambrose","year":"1941","unstructured":"Ambrose, W.: Representation of ergodic flows. 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