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Family of piecewise expanding maps having singular measure as a limit of ACIMs

Published online by Cambridge University Press:  28 November 2011

ZHENYANG LI ,
PAWEŁ GÓRA ,
ABRAHAM BOYARSKY ,
HARALD PROPPE  and
PEYMAN ESLAMI
ZHENYANG LI
Affiliation:
Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8, Canada (email: zhenyangemail@gmail.com, pgora@mathstat.concordia.ca, boyar@alcor.concordia.ca, proppe@alcor.concordia.ca, peyman_eslami@yahoo.com)
PAWEŁ GÓRA
Affiliation:
Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8, Canada (email: zhenyangemail@gmail.com, pgora@mathstat.concordia.ca, boyar@alcor.concordia.ca, proppe@alcor.concordia.ca, peyman_eslami@yahoo.com)
ABRAHAM BOYARSKY
Affiliation:
Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8, Canada (email: zhenyangemail@gmail.com, pgora@mathstat.concordia.ca, boyar@alcor.concordia.ca, proppe@alcor.concordia.ca, peyman_eslami@yahoo.com)
HARALD PROPPE
Affiliation:
Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8, Canada (email: zhenyangemail@gmail.com, pgora@mathstat.concordia.ca, boyar@alcor.concordia.ca, proppe@alcor.concordia.ca, peyman_eslami@yahoo.com)
PEYMAN ESLAMI
Affiliation:
Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8, Canada (email: zhenyangemail@gmail.com, pgora@mathstat.concordia.ca, boyar@alcor.concordia.ca, proppe@alcor.concordia.ca, peyman_eslami@yahoo.com)
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Abstract

Keller [Stochastic stability in some chaotic dynamical systems. Monatsh. Math.94(4) (1982), 313–333] introduced families of W-shaped maps that can have a great variety of behaviors. As a family approaches a limit W map, he observed behavior that was either described by a probability density function (PDF) or by a singular point measure. Based on this, Keller conjectured that instability of the absolutely continuous invariant measure (ACIM) can result only from the existence of small invariant neighborhoods of the fixed critical point of the limit map. In this paper, we show that the conjecture is not true. We construct a very simple family of W-maps with ACIMs supported on the whole interval, whose limiting dynamical behavior is captured by a singular measure. Key to the analysis is the use of a general formula for invariant densities of piecewise linear and expanding maps [P. Góra. Invariant densities for piecewise linear maps of interval. Ergod. Th. & Dynam. Sys. 29(5) (2009), 1549–1583].

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 1 , February 2013 , pp. 158 - 167
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Baladi, V. and Smania, D.. Alternative proofs of linear response for piecewise expanding unimodal maps. Ergod. Th. & Dynam. Sys. 30 (2010), 1–20.CrossRefGoogle Scholar
[2]Boyarsky, A. and Góra, P.. Laws of Chaos. Invariant Measures and Dynamical Systems in One Dimension (Probability and its Applications). Birkhäuser, Boston, MA, 1997.Google Scholar
[3]Dellnitz, M., Froyland, G. and Seertl, S.. On the isolated spectrum of the Perron–Frobenius operator. Nonlinearaity 13 (2000), 1171–1188.CrossRefGoogle Scholar
[4]Dunford, N. and Schwartz, J. T.. Linear Operators, Part I: General Theory. Interscience Publ. Inc., NY, 1964.Google Scholar
[5]Eslami, P. and Misiurewicz, M.. Singular limits of absolutely continuous invariant measures for families of transitive maps. J. Difference Equ. Appl., doi:10.1080/10236198.2011.590480.Google Scholar
[6]Góra, P.. Invariant densities for piecewise linear maps of interval. Ergod. Th. & Dynam. Sys. 29(5) (2009), 1549–1583.CrossRefGoogle Scholar
[7]Góra, P.. On small stochastic perturbations of one-sided subshift of finite type. Bull. Acad. Polon. Sci. 27 (1979), 47–51.Google Scholar
[8]Góra, P. and Boyarsky, A.. Absolutely continuous invariant measures for piecewise expanding C 2 transformations in R N. Israel J. Math. 67 (1989), 272–286.CrossRefGoogle Scholar
[9]Keller, G.. Stochastic stability in some chaotic dynamical systems. Monatsh. Math. 94(4) (1982), 313–333.CrossRefGoogle Scholar
[10]Keller, G. and Liverani, C.. Stability of the spectrum for transfer operators. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 28(1) (1999), 141–152.Google Scholar
[11]Kowalski, Z. S.. Invariant measures for piecewise monotonic transformation has a positive lower bound on its support. Bull. Acad. Polon. Sci. Sér. Sci. Math. 27(1) (1979), 53–57.Google Scholar
[12]Li, T.-Y. and Yorke, J. A.. Ergodic transformations from an interval into itself. Trans. Amer. Math. Soc. 235 (1978), 183–192.CrossRefGoogle Scholar
[13]Murray, R.. Approximation of invariant measures for a class of maps with indifferent fixed points, University of Waikato, Mathematics Research Report Series II No. 106. (2005) http://www.math.canterbury.ac.nz/∼r.murray/files/ulamifpnum.pdf.Google Scholar
[14]Tokman, C. G., Hunt, B. R. and Wright, P.. Approximating invariant densities of metastable systems. Ergod. Th. & Dynam. Sys. 31 (2011), 1345–1361.CrossRefGoogle Scholar