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Properties of topologically transitive maps on the real line. (English) Zbl 1015.37030

Topologically transitive maps defined on intervals of the real line are studied. Let \(I\) be any interval of the real line \(\mathbb{R}\). \(f:I\to I\) is topologically transitive in \(I\) if for any nonempty open subsets \(U, W\) of \(I\) there is a positive integer \(n\) such that \(f^n (U) \cap W\) is nonempty. A point \(x\in I\) is critical point of \(f\) if in every neighborhood \(V\) of \(x\) there are \(y,z \in V\) such that \(f(y)=f(z)\). Let \(C\) be the set of all critical points of \(f\). It is shown that for any topologically transitive map on \(\mathbb{R}\) the sets \(C\) and \(f(C)\) are unbounded, there are at most three open and invariant sets of \(f\) and with a single possible exception for any \(x\in \mathbb{R}\) the backward orbit of \(x\) is dense in \(\mathbb{R}\).

MSC:

37E05 Dynamical systems involving maps of the interval
54H20 Topological dynamics (MSC2010)
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