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The second iterate of a map with dense orbit. (English) Zbl 0853.54036

Summary: Suppose that \(X\) is a Hausdorff topological space having no isolated points and that \(f: X\to X\) is continuous. We show that if the orbit of a point \(x\in X\) under \(f\) is dense in \(X\) while the orbit of \(x\) under \(f\circ f\) is not, then the space \(X\) decomposes into three sets relative to which the dynamics of \(f\) are easy to describe. This decomposition has the following consequence: suppose that \(x\) has dense orbit under \(f\) and that the closure of the set of points of \(X\) having odd period under \(f\) has nonempty interior; then \(x\) has dense orbit under \(f\circ f\).

MSC:

54H20 Topological dynamics (MSC2010)
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
47A15 Invariant subspaces of linear operators

Keywords:

dense orbit
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References:

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