Delays, recurrence and ordinals (Q2766392)

Given a Polish space \(X\) and a continuous function \(f : X \to X\), say that \(x \in X\) attacks \(y \in X\) if \(y = \lim_{n \to \infty} f^{\alpha (n)} (x)\) for some strictly increasing function \(\alpha : \omega \to \omega\). \(\omega_f (x)\) 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 \(\lambda\). The score \(\theta (x,f)\) is the least ordinal \(\theta\) with \(A^\theta (x,f) = A^{\theta + 1} (x,f)\). \(A (x,f) = A^{\theta (x,f)} (x,f)\) is the abode, and \(E(x,f) = \omega_f (x) \setminus A(x,f)\) is the escape set. NEWLINENEWLINENEWLINEThe author shows that \(\theta (x,f)\) is at most the first uncountable ordinal \(\omega_1\), and that it is countable in case \(A (x,f)\) is a Borel set. This is proved by characterizing membership in \(E(x,f)\) via wellfoundedness of a certain tree and then appealing to the classical boundedness theorem in descriptive set theory. NEWLINENEWLINENEWLINESay that \(z \in X\) is a recurrent point if it attacks itself. In general, recurrent points need not exist, but if \(X\) is compact, each \(x \in X\) attacks at least one recurrent \(z \in X\). Furthermore, \(y \in A(x,f)\) if and only if there is a recurrent \(z\) which attacks \(y\) and is attacked by \(x\). Therefore, recurrent points exist if and only if \(A(x,f) \neq \emptyset\). NEWLINENEWLINENEWLINEThe shift function \(s: \omega^\omega \to \omega^\omega\) on the Baire space \(\omega^\omega\) is defined by \(s (y) (n) = y (n+1)\). Refining the rank analysis which permeates many of the proofs, the author shows that for each countable ordinal \(\zeta\) there is \(x \in \omega^\omega\) with \(\theta (x,s) = \zeta\). By an even more sophisticated argument, he obtains the analogous result for a compact space. Finally, a connection to effective descriptive set theory is given, and a new complete analytic set is constructed.