Invariant measures for piecewise convex transformations of an interval

Tom 152 / 2002

Christopher Bose, Véronique Maume-Deschamps, Bernard Schmitt, S. Sujin Shin Studia Mathematica 152 (2002), 263-297 MSC: Primary 37E05, 37A05, 37A25; Secondary 37A40. DOI: 10.4064/sm152-3-5

Streszczenie

We investigate the existence and ergodic properties of absolutely continuous invariant measures for a class of piecewise monotone and convex self-maps of the unit interval. Our assumption entails a type of average convexity which strictly generalizes the case of individual branches being convex, as investigated by Lasota and Yorke (1982). Along with existence, we identify tractable conditions for the invariant measure to be unique and such that the system has exponential decay of correlations on bounded variation functions and Bernoulli natural extension. In the case when there is more than one invariant density we identify a dominant component over which the above properties also hold. Of particular note in our investigation is the lack of smoothness or uniform expansiveness assumptions on the map or its powers.

Autorzy

  • Christopher BoseDepartment of Mathematics and Statistics
    University of Victoria
    P.O. Box 3045
    Victoria, B.C., V8W 3P4, Canada
    e-mail
  • Véronique Maume-DeschampsUMR 5584 du CNRS
    Laboratoire de Topologie
    Université de Bourgogne
    B.P. 400
    21011 Dijon Cedex, France
    e-mail
  • Bernard SchmittUMR 5584 du CNRS
    Laboratoire de Topologie
    Université de Bourgogne
    B.P. 400
    21011 Dijon Cedex, France
    e-mail
  • S. Sujin ShinDepartment of Mathematics
    Korea Advanced Institute
    of Science and Technology
    373-1, Guseong-dong, Yuseong-gu
    Daejon, 305-701, South Korea
    e-mail

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