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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)
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