{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T14:56:08Z","timestamp":1772290568910,"version":"3.50.1"},"reference-count":54,"publisher":"American Mathematical Society (AMS)","issue":"288","license":[{"start":{"date-parts":[[2014,11,12]],"date-time":"2014-11-12T00:00:00Z","timestamp":1415750400000},"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":["Math. Comp."],"abstract":"<p>We consider a generalisation of Ulam\u2019s method for approximating invariant densities of one-dimensional maps. Rather than use piecewise constant polynomials to approximate the density, we use polynomials of degree <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n\">\n  <mml:semantics>\n    <mml:mi>n<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">n<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> which are defined by the requirement that they preserve the measure on <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n plus 1\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>n<\/mml:mi>\n      <mml:mo>+<\/mml:mo>\n      <mml:mn>1<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">n+1<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> neighbouring subintervals. Over the whole interval, this results in a discontinuous piecewise polynomial approximation to the density. We prove error results where this approach is used to approximate smooth densities. We also consider the computation of the Lyapunov exponent using the polynomial density and show that the order of convergence is one order better than for the density itself. Together with using cubic polynomials in the density approximation, this yields a very efficient method for computing highly accurate estimates of the Lyapunov exponent. We illustrate the theoretical findings with some examples.<\/p>","DOI":"10.1090\/s0025-5718-2013-02811-6","type":"journal-article","created":{"date-parts":[[2013,11,12]],"date-time":"2013-11-12T19:52:42Z","timestamp":1384285962000},"page":"1869-1902","source":"Crossref","is-referenced-by-count":1,"title":["Computing the invariant measure and the Lyapunov exponent for one-dimensional maps using a measure-preserving polynomial basis"],"prefix":"10.1090","volume":"83","author":[{"given":"Philip","family":"Aston","sequence":"first","affiliation":[]},{"given":"Oliver","family":"Junge","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2013,11,12]]},"reference":[{"issue":"5","key":"1","doi-asserted-by":"publisher","first-page":"1428","DOI":"10.1016\/j.aim.2009.03.004","article-title":"Invariant measures for interval maps with critical points and singularities","volume":"221","author":"Ara\u00fajo, V\u00edtor","year":"2009","journal-title":"Adv. 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