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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
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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}$.


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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

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