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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Absolutely continuous S.R.B. measures for random Lasota-Yorke maps
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by Jérôme Buzzi PDF
Trans. Amer. Math. Soc. 352 (2000), 3289-3303 Request permission

Abstract:

A. Lasota and J. A. Yorke proved that a piecewise expanding interval map admits finitely many ergodic absolutely continuous invariant probability measures. We generalize this to the random composition of such maps under conditions which are natural and less restrictive than those previously studied by Morita and Pelikan. For instance our conditions are satisfied in the case of arbitrary random $\beta$-transformations, i.e., $x\mapsto \beta x\mod 1$ on $[0,1]$ where $\beta$ is chosen according to any stationary stochastic process (in particular, not necessarily i.i.d.) with values in $]1,\infty [$. Résumé. A. Lasota et J. A. Yorke ont montré qu’une application de l’intervalle dilatante par morceaux admet un nombre fini de mesures de probabilité invariantes et ergodiques absolument continues. Nous généralisons ce résultat à la composition aléatoire de telles applications sous des conditions naturelles, moins restrictives que celles précédemment envisagées par Morita et Pelikan. Par exemple, nos conditions sont satisfaites par toute $\beta$-transformation aléatoire, i.e., $x\mapsto \beta x\mod 1$ sur $[0,1]$ avec $\beta$ choisi selon un processus stochastique stationnaire quelconque (en particulier, non-nécessairement i.i.d.) à valeurs dans $]1,\infty [$.
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Additional Information
  • Jérôme Buzzi
  • Affiliation: Institut de Mathématiques de Luminy, 163 av. de Luminy, Case 907, 13288 Marseille Cedex 9, France
  • Address at time of publication: Centre de Mathématiques, Ecole Polytechnique, 91128 Palaiseau Cedex, France
  • Email: buzzi@iml.univ-mrs.fr
  • Received by editor(s): January 1, 2500
  • Received by editor(s) in revised form: January 1, 1998
  • Published electronically: March 24, 2000
  • Additional Notes: Work partly done at the Laboratoire de Topologie de Dijon, Université de Bourgogne, France
  • © Copyright 2000 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 352 (2000), 3289-3303
  • MSC (2000): Primary 37A25, 37H99
  • DOI: https://doi.org/10.1090/S0002-9947-00-02607-6
  • MathSciNet review: 1707698