## A scenario for transferring high scores.(English)Zbl 1073.03027

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 $$x$$. In analogy to the Cantor-Bendixson derivative, define recursively a decreasing sequence of sets by $$A^0 (x,f) = \omega_f (x)$$, $$A^{\beta + 1} (x,f) = \bigcup \{ \omega_f (y) : y \in A^\beta (x,f) \}$$ and $$A^\lambda (x,f) = \bigcap_{\beta < \lambda} A^\beta (x,f)$$ for limit ordinals
In Proc. Lond. Math. Soc., III. Ser. 82, No. 2, 257–298 (2001; Zbl 1017.03025) the author proved that $$\theta (x,f)$$ is at most the first uncountable ordinal $$\omega_1$$, and in Math. Proc. Camb. Philos. Soc. 138, No. 3, 465–485 (2005; Zbl 1084.03040) he showed that if $$s$$ is the shift function on the Baire space $$\omega^\omega$$ defined by $$s (y) (n) = y (n+1)$$ there is a point $$x$$ with $$\theta (x,s) = \omega_1$$.
The author discusses the question of what other spaces $$X$$ and functions $$f$$ admit the existence of a point of uncountable score, and shows that if there is a continuous surjection $$\Psi$$ from $$X$$ onto either the Baire space or the Cantor space with $$\Psi (f(x)) = s (\Psi (x))$$ satisfying several additional properties, then $$\theta (x,f) = \theta (\Psi (x), s)$$ for all $$x \in X$$ so that there are points in $$X$$ of uncountable score.

### 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)

### Citations:

Zbl 1017.03025; Zbl 1084.03040
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