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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Exact dynamical systems and the Frobenius-Perron operator


Authors: A. Lasota and James A. Yorke
Journal: Trans. Amer. Math. Soc. 273 (1982), 375-384
MSC: Primary 28D05; Secondary 58F20
DOI: https://doi.org/10.1090/S0002-9947-1982-0664049-X
MathSciNet review: 664049
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Abstract: Conditions are investigated which guarantee exactness for measurable maps on measure spaces. The main application is to certain piecewise continuous maps $ T$ on $ [0,1]$ for which $ T'(0) > 1$. We assume $ [0,1]$ can be broken into intervals on which $ T$ is continuous and convex and at the left end of these intervals $ T = 0$ and $ dt/dx > 0$. Such maps have an invariant absolutely continuous density which is exact.


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  • [1] É. Borel, Les probabilités denombrables et leurs aplications aritmétiques, Rend. Circ. Mat. Palermo 27 (1909), 247-271.
  • [2] A. O. Gelfond, A common property of number systems, Izv. Akad. Nauk SSSR Ser. Mat. 23 (1959), 809-814. (Russian) MR 0109817 (22:702)
  • [3] A. Lasota, Invariant measures and functional equations, Aequationes Math. 9 (1973), 193-200. MR 0328026 (48:6368)
  • [4] -, A fixed point theorem and its application in ergodic theory, Tôhoku Math. J. 32 (1980), 567-575. MR 601927 (82c:47071)
  • [5] M. Lin, Mixing for Markov operators, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 16 (1971), 231-242. MR 0309207 (46:8317)
  • [6] W. Parry, On the $ \beta $-expansion of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1970), 401-416. MR 0142719 (26:288)
  • [7] A. Rényi, Representation for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477-493. MR 0097374 (20:3843)
  • [8] V. A. Rochlin, Exact endomorphisms of Lebesgue spaces, Izv. Akad. Nauk SSSR Ser. Mat. 25 (1971), 499-530; Amer. Math. Soc. Transl. (2) 39 (1964), 1-36.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0664049-X
Article copyright: © Copyright 1982 American Mathematical Society

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