1Department of Mathematics and Statistics University of Victoria P.O. Box 3045 Victoria, B.C., V8W 3P4, Canada 2UMR 5584 du CNRS Laboratoire de Topologie Université de Bourgogne B.P. 400 21011 Dijon Cedex, France 3Department 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
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.
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
Affiliations des auteurs :
Christopher Bose
1
;
Véronique Maume-Deschamps
2
;
Bernard Schmitt
2
;
S. Sujin Shin
3
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
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author = {Christopher Bose and V\'eronique Maume-Deschamps and Bernard Schmitt and S. Sujin Shin},
title = {Invariant measures for piecewise convex
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journal = {Studia Mathematica},
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year = {2002},
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AU - Christopher Bose
AU - Véronique Maume-Deschamps
AU - Bernard Schmitt
AU - S. Sujin Shin
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Christopher Bose; Véronique Maume-Deschamps; Bernard Schmitt; S. Sujin Shin. Invariant measures for piecewise convex
transformations of an interval. Studia Mathematica, Tome 152 (2002) no. 3, pp. 263-297. doi: 10.4064/sm152-3-5
