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Frequently hypercyclic operators. (English) Zbl 1115.47005

The authors investigate the subject of linear dynamics by studying the notion of frequent hypercyclicity for bounded operators \(T\) on separable complex \({\mathcal F}\)-spaces: \(T\) is frequently hypercyclic if there exists a vector \(x\) such that for every nonempty open subset \(U\) of \(X\), the set of integers \(n\) such that \(T^nx\) belongs to \(U\) has positive lower density. They give several criteria for frequent hypercyclicity, and this leads them in particular to study linear transformations from the point of view of ergodic theory. Several other topics which are classical in hypercyclicity theory are also investigated in the frequent hypercyclicity setting.

MSC:

47A16 Cyclic vectors, hypercyclic and chaotic operators
47A35 Ergodic theory of linear operators
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37A05 Dynamical aspects of measure-preserving transformations
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