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Compactness of trajectories of dynamical systems in complete uniform spaces. (English) Zbl 0615.54029

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

MSC:

54H20 Topological dynamics (MSC2010)
28D05 Measure-preserving transformations
54H15 Transformation groups and semigroups (topological aspects)
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