Mean Convergence of Periodic Pseudotrajectories and Invariant Measures of Dynamical Systems

Abstract

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.