Benaïm, Michel; Schreiber, Sebastian J. Ergodic properties of weak asymptotic pseudotrajectories for semiflows. (English) Zbl 0998.37013 J. Dyn. Differ. Equations 12, No. 3, 579-598 (2000). The first author, with 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 \(X\). Reviewer: Athanase Papadopoulos (Strasbourg) Cited in 9 Documents MSC: 37H99 Random dynamical systems 37C50 Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics 37A10 Dynamical systems involving one-parameter continuous families of measure-preserving transformations 37M25 Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.) 60G48 Generalizations of martingales Keywords:asymptotic autonomous semiflows; stochastic algorithms; stochastic differential equations; invariant measures; minimal center of attraction Citations:Zbl 0878.58053 PDFBibTeX XMLCite \textit{M. Benaïm} and \textit{S. J. Schreiber}, J. Dyn. Differ. Equations 12, No. 3, 579--598 (2000; Zbl 0998.37013) Full Text: DOI