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Transactions of the American Mathematical Society

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

 

 

Absolutely continuous S.R.B. measures for random Lasota-Yorke maps


Author: Jérôme Buzzi
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
Published electronically: March 24, 2000
MathSciNet review: 1707698
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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 $]1,\infty [$.


RSESUM´E. 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

DOI: https://doi.org/10.1090/S0002-9947-00-02607-6
Keywords: Absolutely continuous invariant measures; piecewise monotonic maps; random maps; transfer operator; bounded variation
Received by editor(s): January 1, 2025
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
Article copyright: © Copyright 2000 American Mathematical Society