Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

Asymptotic periodicity of the iterates of Markov operators


Authors: A. Lasota, T.-Y. Li and J. A. Yorke
Journal: Trans. Amer. Math. Soc. 286 (1984), 751-764
MSC: Primary 47A35; Secondary 28D05, 58F11, 82A40
DOI: https://doi.org/10.1090/S0002-9947-1984-0760984-4
MathSciNet review: 760984
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We say $ P:{L^1} \to {L^1}$ is a Markov operator if (i) $ Pf \geq 0$ for $ f \geq 0$ and (ii) $ \Vert Pf\Vert = \Vert f\Vert $ if $ f \geq 0$. It is shown that any Markov operator $ P$ has certain spectral decomposition if, for any $ f \in {L^1}$ with $ f \geq 0$ and $ \Vert f\Vert = 1$, $ {P^n}f \to \mathcal{F}$ when $ n \to \infty $, where $ \mathcal{F}$ is a strongly compact subset of $ {L^1}$. It follows from this decomposition that $ {P^n}f$ is asymptotically periodic for any $ f \in {L^1}$.


References [Enhancements On Off] (What's this?)

  • [1] N. Dunford and J. T. Schwartz, Linear operators. $ 1$, Interscience, New York, 1958.
  • [2] S. R. Foguel, The ergodic theory of Markov processes, Van Nostrand Reinhold, New York, 1969. MR 0261686 (41:6299)
  • [3] F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z. 180 (1982), 119-140. MR 656227 (83h:28028)
  • [4] E. Hopf, The general temporally discrete Markov process, J. Rational Mech. Anal. 3 (1959), 13-45. MR 0060181 (15:636b)
  • [5] C. T. Ionescu Tulcea and G. Marinescu, Théorie ergodigue pout des classes d'opérations non completement continues, Ann. of Math. (2) 52 (1950), 140-147. MR 0037469 (12:266g)
  • [6] B. Jamison, Ergodic decompositions induced by certain Markov operators, Trans. Amer. Math. Soc. 117 (1965), 451-468. MR 0207041 (34:6857)
  • [7] G. Keller, Ergodicité et mesures invariantes pour les transformations dilatantes par morceaux d'une région bornée du plan, C. R. Acad. Sci. Paris Sér. A 289 (1979), 625-627. MR 556443 (80k:28016)
  • [8] M. A. Krasnosel'skii, Integral operators in spaces of summable functions, Noordhoff, Groningen, 1976.
  • [9] A. Lasota, Statistical stability of deterministic systems, Lecture Notes in Math., vol. 1017, Springer-Verlag, Berlin and New York, 1983, pp. 386-419. MR 726599 (85d:28012)
  • [10] A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481-488. MR 0335758 (49:538)
  • [11] -, Exact dynamical systems and the Frobenius-Perron operator, Trans. Amer. Math. Soc. 273 (1982), 375-384. MR 664049 (84d:28023)
  • [12] M. Lin and R. Sine, A spectral condition for strong convergence of Markov operators, Z. Wahrsch. Verw. Gebiete 29 (1979), 27-29. MR 521529 (80d:47010)
  • [13] M. Misiurewicz, Absolutely continuous measures for certain maps of an interval, Inst. Hautes Étude Sci. Publ. Math. 53 (1981), 17-51. MR 623533 (83j:58072)
  • [14] H. H. Schaefer, On positive contractions in $ {L^p}$ spaces, Trans. Amer. Math. Soc. 257 (1980), 261-268. MR 549167 (81b:47048)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47A35, 28D05, 58F11, 82A40

Retrieve articles in all journals with MSC: 47A35, 28D05, 58F11, 82A40


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0760984-4
Keywords: Markov operator, invariant measures, spectral decomposition, asymptotic periodicity
Article copyright: © Copyright 1984 American Mathematical Society

American Mathematical Society