Exact dynamical systems and the Frobenius-Perron operator
HTML articles powered by AMS MathViewer
- by A. Lasota and James A. Yorke PDF
- Trans. Amer. Math. Soc. 273 (1982), 375-384 Request permission
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.References
-
É. Borel, Les probabilités denombrables et leurs aplications aritmétiques, Rend. Circ. Mat. Palermo 27 (1909), 247-271.
- A. O. Gel′fond, A common property of number systems, Izv. Akad. Nauk SSSR. Ser. Mat. 23 (1959), 809–814 (Russian). MR 0109817
- A. Lasota, Invariant measures and functional equations, Aequationes Math. 9 (1973), 193–200. MR 328026, DOI 10.1007/BF01832626
- Andrzej Lasota, A fixed point theorem and its application in ergodic theory, Tohoku Math. J. (2) 32 (1980), no. 4, 567–575. MR 601927, DOI 10.2748/tmj/1178229541
- Michael Lin, Mixing for Markov operators, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 19 (1971), 231–242. MR 309207, DOI 10.1007/BF00534111
- W. Parry, On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416 (English, with Russian summary). MR 142719, DOI 10.1007/BF02020954
- A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493. MR 97374, DOI 10.1007/BF02020331 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.
Additional Information
- © Copyright 1982 American Mathematical Society
- 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