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Transactions of the American Mathematical Society

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On the existence of invariant measures for piecewise monotonic transformations

Authors: A. Lasota and James A. Yorke
Journal: Trans. Amer. Math. Soc. 186 (1973), 481-488
MSC: Primary 28A70
MathSciNet review: 0335758
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Abstract: A class of piecewise continuous, piecewise $ {C^1}$ transformations on the interval [0, 1] is shown to have absolutely continuous invariant measures.

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  • [1] Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR 0117523
  • [2] A. O. Gel′fond, A common property of number systems, Izv. Akad. Nauk SSSR. Ser. Mat. 23 (1959), 809–814 (Russian). MR 0109817
  • [3] A. Lasota, Invariant measures and functional equations, Aequationes Math. 9 (1973), 193–200. MR 0328026
  • [4] W. Parry, On the 𝛽-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416 (English, with Russian summary). MR 0142719
  • [5] A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar 8 (1957), 477–493. MR 0097374
  • [6] V. A. Rohlin, Exact endomorphisms of a Lebesgue space, Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 499–530 (Russian). MR 0143873
  • [7] S. M. Ulam, A collection of mathematical problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York-London, 1960. MR 0120127
  • [8] Michael S. Waterman, Some ergodic properties of multi-dimensional 𝑓-expansions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 16 (1970), 77–103. MR 0282939
  • [9] André Avez, Propriétés ergodiques des endomorphisms dilatants des variétés compactes, C. R. Acad. Sci. Paris Sér. A-B 266 (1968), A610–A612 (French). MR 0231389
  • [10] K. Krzyżewski and W. Szlenk, On invariant measures for expanding differentiable mappings, Studia Math. 33 (1969), 83–92. MR 0245761

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Keywords: Frobenius-Perron operator, invariant measures
Article copyright: © Copyright 1973 American Mathematical Society