## 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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