Ergodic properties of weak asymptotic pseudotrajectories for semiflows
The first author, with \textit{M. W. Hirsch} [J. Dyn. Differ. Equ. 8, 141-176 (1996; Zbl 0878.58053)], studied the limiting behavior of asymptotic pseudotrajectories for a semiflow, which they defined as follows. If \(\Phi\) is a semiflow on a metric space, then \(\{ X(t):t\geq 0\}\) is an asymptotic pseudotrajectory for \(\Phi\) if for any \(T>0\), one has \[ \lim_{t\to\infty} \sup_{0\leq h\leq T} d(X(t+h),\Phi_h(X(t)))=0. \] This theory provides a general framework for the study of long-term behavior of a large class of nonautonomous systems of difference/differential equations. In this paper, the authors are concerned with the ergodic and statistical behavior of asymptotic trajectories, and in fact in a class of more general statistical processes \(X(t)\) on the metric space which they call weak asymptotic pseudotrajectories. The main results are: (1) the weak\(^*\) limit points of the empirical measures for \(X\) are almost surely \(\Phi\)-invariant measures; (2) for any semiflow \(\Phi\), there exists a weak asymptotic pseudotrajectory of \(\Phi\) such that the set of weak\(^*\) limit points of its empirical measures is almost surely equal to the set of ergodic measures for \(\Phi\); (3) given an asymptotic pseudotrajectory \(X\) for a semiflow \(\Phi\), then they derive conditions on \(\Phi\) that insure convergence of the empirical measures of \(X\).
scientific article; zbMATH DE number 1556551