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The point spectrum of Frobenius-Perron
and Koopman operators


Author: J. Ding
Journal: Proc. Amer. Math. Soc. 126 (1998), 1355-1361
MSC (1991): Primary 47A35, 28D05, 47A10, 47B38
DOI: https://doi.org/10.1090/S0002-9939-98-04188-4
MathSciNet review: 1443148
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Abstract: We present some results on the point spectrum of the Frobenius-Perron operator $P:L^1\to L^1$ and the Koopman operator $U\colon L^\infty\to L^\infty$ associated with a nonsingular transformation $S\colon X\to X$ on a $\sigma$-finite measure space $(X,\Sigma,\mu)$.


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  • 1. S. Albeverio and R. Hoegh-Krohn, Frobenius-theory for positive maps of von Neumann algebras, Comm. Math. Phys. 64 (1978), 83-94. MR 81m:46091
  • 2. C. Beck and F. Schlögl, Thermodynamics of Chaotic Systems, Cambridge University Press, 1993. MR 94m:58134
  • 3. C. Chiu, Q. Du and T.-Y. Li, Error estimates of the Markov finite approximation of the Frobenius-Perron operator, Nonlinear Anal. TMA 19 (4) (1992), 291-308. MR 93f:28014
  • 4. J.Ding, Decomposition theorems for Koopman operators, Nonlinear Anal. TMA 28 (1997), 1011-1018. CMP 97:05
  • 5. J. Ding, Q. Du and T.-Y. Li, The spectral analysis of Frobenius-Perron operators, J. Math. Anal. Appl. 184 (2) (1994), 285-301. MR 95c:47007
  • 6. J. Ding, Q. Du and T.-Y. Li, High order approximation of the Frobenius-Perron operator, Applied Math. Comput. 53 (1993), 151-171. MR 94d:65036
  • 7. J. Ding and W. E. Hornor, A new approach to Frobenius-Perron operators, J. Math. Anal. Appl., 187 (3) (1994), 1047-1058. MR 95k:28033
  • 8. J. Ding and T.-Y. Li, A convergence rate analysis for Markov finite approximations to a class of Frobenius-Perron operators, to appear: Nonlinear Anal. TMA.
  • 9. N. Dunford and J. Schwartz, Linear Operators, Part I, General Theory, Interscience, 1958. MR 22:8302
  • 10. D. E. Evans and R. Hoegh-Krohn, Spectral properties of positive maps on $C^*$-algebras, J. London Math. Soc. 17 (2) (1978), 345-355. MR 58:2319
  • 11. M. Fannes, B. Nachtergaele, and R. F. Werner, Finitely correlated states on quantum spin chains, Comm. Math. Phys. 144 (1992), 443-490. MR 93i:82006
  • 12. S. R. Foguel, The Ergodic Theory of Markov Processes, Van Nostrand Reinhold, 1969. MR 41:6299
  • 13. F. Y. Hunt and W. M. Miller, On the approximation of invariant measures, J. Stat. Phys. 66 (1992), 535-548. MR 93a:58105
  • 14. A. Lasota and M. Mackey, Chaos, Fractals, and Noises, Second Ed., Springer-Verlag, 1994. MR 94j:58102
  • 15. T. Y. Li, Finite approximation for the Frobenius-Perron operator, a solution to Ulam's conjecture, J. Approx. Theory 17 (1976), 177-186. MR 54:811
  • 16. H. H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag, 1974. MR 54:11023
  • 17. Ya. G. Sinai, Topics in Ergodic Theory, Princeton University Press, 1994. MR 95j:28017
  • 18. S. Ulam, A Collection of Mathematical Problems, Interscience, 1960. MR 22:10884
  • 19. P. Walter, An introduction to Ergodic Theory, Springer-Verlag, 1982. MR 84e:28017

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Additional Information

J. Ding
Affiliation: Department of Mathematics, The University of Southern Mississippi, Hattiesburg, Mississippi 39406-5045
Email: jding@yizhi.st.usm.edu

DOI: https://doi.org/10.1090/S0002-9939-98-04188-4
Keywords: Frobenius-Perron operator, Koopman operator
Additional Notes: Research was supported in part by a grant from the Minority Scholars Program through the University of Southern Mississippi
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society

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