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Choosing an attacker by a local derivation. (English) Zbl 1073.03026

For a Polish space \((X,d)\) and a continuous function \(f : X \to X\), say that \(x \in X\) attacks \(y \in X\) if for all positive integers \(m\) there is \(\ell \geq m\) with \(d (f^\ell (x) , y) < {1 \over m}\). \(\omega_f (x)\) is the set of points attacked by \(x\). \(x \) is a recurrent point if it attacks itself.
For a fixed Polish space \(X\), continuous function \(f\) and point \(c \in X\), the author provides a uniform way for choosing for each \(y\) a point \(x\) such that \(c\) attacks \(x\) and \(x\) attacks \(y\), provided there is such an \(x\). This result is obtained by a Cantor-Bendixson-style rank analysis of \(\omega_f (c)\) and methods from classical descriptive set theory. By refining this method, he also describes a uniform way for choosing a recurrent \(x\) such that \(c\) attacks \(x\) and \(x\) attacks \(y\), provided there is such a recurrent \(x\).

MSC:

03E15 Descriptive set theory
37B10 Symbolic dynamics
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
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