Bartoszek, Wojciech; Downarowicz, Tomasz Compactness of trajectories of dynamical systems in complete uniform spaces. (English) Zbl 0615.54029 Rend. Circ. Mat. Palermo, II. Ser. Suppl. 10, 13-16 (1985). Let \(\{\phi_ t\}\), \(t\geq 0\), be an equicontinuous semigroup of mappings acting on a complete uniform space X. Let \(X_ 0\) denote the set of all \(x\in X\) such that the trajectory of x is precompact and let \(\omega\) (x) be the omega limit set of \(x\in X\). The following conditions are shown to be equivalent: (a) \(x\in X_ 0\), (b) \(\omega\) (x) is nonempty and compact, (c) there exists a \(\phi_ t\)-invariant probability measure \(\mu_ x\) on \(\omega\) (x), (d) for every continuous function \(F: X\to E\) (a Banach space) the Bochner integrals \(T^{-1}\int^{T}_{0}F(\phi_ t(x))dt\) are convergent as \(T\to \infty\) to a limit \(\bar F(E)\in E\). If the above holds, then \(\bar F\) is a continuous invariant function on \(X_ 0\) and equals \(\int_{\omega (x)}F(y)\mu_ x(dy)\) where \(\mu_ x\) is the unique invariant probability measure on \(\omega(x)\). Reviewer: W.R.Utz Cited in 1 Document MSC: 54H20 Topological dynamics (MSC2010) 28D05 Measure-preserving transformations 54H15 Transformation groups and semigroups (topological aspects) Keywords:asymptotic behaviour of trajectory; equicontinuous semigroup of mappings; complete uniform space; invariant probability measure; Bochner integrals PDFBibTeX XMLCite \textit{W. Bartoszek} and \textit{T. Downarowicz}, Suppl. Rend. Circ. Mat. Palermo (2) 10, 13--16 (1985; Zbl 0615.54029)