Invariant measures for piecewise convex transformations of an interval

Christopher Bose¹ ; Véronique Maume-Deschamps² ; Bernard Schmitt² ; S. Sujin Shin³

¹ Department of Mathematics and Statistics University of Victoria P.O. Box 3045 Victoria, B.C., V8W 3P4, Canada
² UMR 5584 du CNRS Laboratoire de Topologie Université de Bourgogne B.P. 400 21011 Dijon Cedex, France
³ Department of Mathematics Korea Advanced Institute of Science and Technology 373-1, Guseong-dong, Yuseong-gu Daejon, 305-701, South Korea

Studia Mathematica, Tome 152 (2002) no. 3, pp. 263-297

Résumé

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.

Détail

DOI : 10.4064/sm152-3-5

Keywords: investigate existence ergodic properties absolutely continuous invariant measures class piecewise monotone convex self maps unit interval assumption entails type average convexity which strictly generalizes individual branches being convex investigated lasota yorke along existence identify tractable conditions invariant measure unique system has exponential decay correlations bounded variation functions bernoulli natural extension there invariant density identify dominant component which above properties particular note investigation lack smoothness uniform expansiveness assumptions map its powers