Geodesic
Mean Convergence of Periodic Pseudotrajectories and Invariant Measures of Dynamical Systems
Matematičeskie zametki, Tome 108 (2020) no. 6, pp. 882-898.

Voir la notice de l'article provenant de la source Math-Net.Ru

A discrete dynamical system generated by a homeomorphism of a compact manifold is considered. A sequence $\omega_n$ of periodic $\varepsilon_n$-trajectories converges in the mean as $\varepsilon_n\to 0$ if, for any continuous function $\varphi$, the mean values on the period $\overline\varphi(\omega_n)$ converge as $n\to\infty$. It is shown that $\omega_n$ converges in the mean if and only if there exists an invariant measure $\mu$ such that $\overline\varphi(\omega_n)$ converges to $\int\varphi\,d\mu$. If a sequence $\omega_n$ converges in the mean and converges uniformly to a trajectory $\operatorname{Tr}$, then the trajectory $\operatorname{Tr}$ is recurrent and its closure is a minimal strictly ergodic set.
Keywords: pseudotrajectory, invariant measure, symbolic image, minimal set, ergodicity. 
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     title = {Mean {Convergence} of {Periodic} {Pseudotrajectories} and {Invariant} {Measures} of {Dynamical} {Systems}},
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G. S. Osipenko. Mean Convergence of Periodic Pseudotrajectories and Invariant Measures of Dynamical Systems. Matematičeskie zametki, Tome 108 (2020) no. 6, pp. 882-898. http://geodesic.mathdoc.fr/item/MZM_2020_108_6_a5/

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