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.


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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