Conceptions of topological transitivity. (English) Zbl 1248.37015
The authors discuss the concept of topological transitivity for a discrete dynamical system generated by a continuous map \(f\) defined on a (nonempty) topological space \(X\). They say that \((X,f)\) is topologically transitive if for every pair of nonempty open sets \(U,V\subset X\) the set \(\{k\in\mathbb{Z}|\;A\cap f^{-k}(B)\not=\emptyset\}\) is not empty. The authors list six further properties related to transitivity and they show that all of them coincide in the case when \(X\) is perfect, Hausdorff, second countable, and non-meager; each of these concepts is traced back in the literature.
Reviewer: Christian Fenske (Gießen)
MSC:
| 37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |
| 54H20 | Topological dynamics (MSC2010) |
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