Abstract
We call a point “dynamically special” if it has a dynamical property, which no other point has. We prove that, for continuous self maps of the real line, all dynamically special points are in the closure of the union of the full orbits of periodic points, critical points and limits at infinity. We completely describe the set of dynamically special points of real polynomial functions. The following characterization for the set of special points is also obtained: A subset of \({\mathbb{R}}\) is the set of dynamically special points for some continuous self map of \({\mathbb{R}}\) if and only if it is closed.
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