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On the Existence of Invariant Measures for Piecewise Monotonic Transformations

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The Theory of Chaotic Attractors

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Abstract

A class of piecewise continuous, piecewise C transforma­tions on the interval [0,1] is shown to have absolutely continuous invariant measures.

(1) The research of both authors was partially supported by the National Science Foundation under grant GP-31386x.

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References

  1. N. Dunford and j. T. Schwartz, Linear operators. I. General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR 22 #8302.

    Google Scholar 

  2. A. O. Gel’fond, A common property of number systems, Izv. Akad. Nauk Ser. Mat. SSSR 23 (1959), 809–814. (Russian) MR 22 702.

    Google Scholar 

  3. A. Lasota, Invariant measures and functional equations, Aequationes Math. (in press).

    Google Scholar 

  4. W. Parry, On the / B-expansion of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416. MR 26 #288.

    Google Scholar 

  5. A. Rényi, Representation for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493. MR 20 #3843.

    Google Scholar 

  6. V. A. Rohlin, Exact endomorphisms of Lebesgue spaces, Izv. Akad. Nauk Ser. Mat. SSSR 25 (1961), 499–530; English transi., Amer. Math. Soc. Transi. (2) 39 (1964), 1–36. MR 26 #1423.

    Google Scholar 

  7. S. M. Ulam, A collection of mathematical problems, Interscience Tracts in Pure and Appl. Math., no. 8, Interscience, New York, 1960. MR 22 #10884.

    Google Scholar 

  8. M. S. Waterman, Some ergodic properties of multidimensional F-expansions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 16 (1970), 77–103. MR 44 # 173.

    Google Scholar 

  9. A. Avez, Propriétés ergodiques des endomorphismes dilatants des variétés compactes C. R. Acad. Sci. Paris Ser. A—B 266 (1968), A610—A612. MR 37 #6944.

    Google Scholar 

  10. K. Krzyzewski and W. Szlenk, On invariant measures for expanding differentiable mappings Studia Math. 33 (1969), 83–92. MR 39 #7067.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Jagellonian University, Cracow, Poland

    A. Lasota & James A. Yorke

  2. Institute for fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Maryland, 20742, USA

    A. Lasota & James A. Yorke

Authors
  1. A. Lasota
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  2. James A. Yorke
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Editor information

Editors and Affiliations

  1. Institute for Physical Science and Technology, University of Maryland, 20742, College Park, MD, USA

    Brian R. Hunt

  2. Department of Mathematics, Michigan State University, 48824, East Lansing, Michigan, USA

    Tien-Yien Li

  3. Department of Mathematics, University of Delaware, 19716, Newark, Delaware, USA

    Judy A. Kennedy

  4. Department of Econometrics, University of Groningen, NL-9700 AV, Groningen, The Netherlands

    Helena E. Nusse

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© 1974 Springer Science+Business Media New York

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Lasota, A., Yorke, J.A. (1974). On the Existence of Invariant Measures for Piecewise Monotonic Transformations. In: Hunt, B.R., Li, TY., Kennedy, J.A., Nusse, H.E. (eds) The Theory of Chaotic Attractors. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21830-4_4

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  • DOI: https://doi.org/10.1007/978-0-387-21830-4_4

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4419-2330-1

  • Online ISBN: 978-0-387-21830-4

  • eBook Packages: Springer Book Archive

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