Abstract
We prove that a Markov operatorT onL 1 has an invariant density if and only if there exists a densityf that satisfies lim sup n→∞‖T n f − f‖ < 2. Using this result, we show that a Frobenius-Perron operatorP is mean ergodic if and only if there exists a densityw such that lim sup n→∞ ‖P n f − w‖<2 for every densityf. Corresponding results hold for strongly continuous semigroups.
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Emel’yanov, E.Y. Invariant densities and mean ergodicity of markov operators. Isr. J. Math. 136, 373–379 (2003). https://doi.org/10.1007/BF02807206
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DOI: https://doi.org/10.1007/BF02807206