Chaotic dynamics of continuous-time topological semi-flows on Polish spaces. (English) Zbl 1320.34069

In the paper under review a semiflow \((\mathbb{R}_+,X,f)\), shortly \((\mathbb{R}_+,X)\), is a continuous monoid action \(f:\mathbb{R}_+\times X\to X\), \((t,x)\mapsto f(t,x)=:tx\) for every \(t\in \mathbb{R}_+, x\in X\), of the additive monoid \(\mathbb{R}_+=[0,\infty)\) on a Polish space \((X,d)\). The main result of the paper is the following theorem: Let \((\mathbb{R}_+,X)\) be a semiflow which is non-minimal, point-transitive and such that the set of points \(x\in X\) whose orbit-closure is a compact minimal subset of \(X\) is dense in \(X\). Then the semiflow \((\mathbb{R}_+,X)\) is sensitive in the following sense: there is a number \(c>0\) such that for every \(x\in X\) there exists a dense \(G_\delta\)-set \(S_c(x)\subset X\) with the property \(\limsup_{t\to\infty} d(tx,ty)\geq c\) for each \(y\in S_c(x)\).
The paper is well written and interesting.


34C28 Complex behavior and chaotic systems of ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
54H20 Topological dynamics (MSC2010)
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