Classical and effective descriptive complexities of -powers

We prove that, for each countable ordinal ξ ≥ 1 , there exist some Σ ξ 0 -complete ω -powers, and some Π ξ 0 -complete ω -powers, extending previous works on the topological complexity of ω -powers [O. Finkel, Topological properties of omega context free languages, Theoretical Computer Science 262 (1–2) (2001) 669–697; O. Finkel, Borel hierarchy and omega context free languages, Theoretical Computer Science 290 (3) (2003) 1385–1405; O. Finkel, An omega-power of a finitary language which is a borel set of infinite rank, Fundamenta informaticae 62 (3–4) (2004) 333–342; D. Lecomte, Sur les ensembles de phrases infinies constructibles a partir d’un dictionnaire sur un alphabet fini, Séminaire d’Initiation a l’Analyse, 1, année 2001–2002; D. Lecomte, Omega-powers and descriptive set theory, Journal of Symbolic Logic 70 (4) (2005) 1210–1232; J. Duparc, O. Finkel, An ω -Power of a Context-Free Language Which Is Borel Above Δ ω 0 , in: S. Bold, B. Löwe, T. Räsch, J. van Benthem (Eds.), in the Proceedings of the international conference foundations of the formal sciences V : Infinite Games, November 26th to 29th, 2004, Bonn, Germany, in: Studies in Logic, vol. 11, College Publications at King’s College, 2007, pp. 109–122]. We prove effective versions of these results; in particular, for each recursive ordinal ξ < ω 1 C K there exist some recursive sets A ⊆ 2 < ω such that A ∞ ∈ Π ξ 0 ∖ Σ ξ 0 (respectively, A ∞ ∈ Σ ξ 0 ∖ Π ξ 0 ), where Π ξ 0 and Σ ξ 0 denote classes of the hyperarithmetical hierarchy. To do this, we prove effective versions of a result by Kuratowski, describing a Π ξ 0 set as the range of a closed subset of the Baire space ω ω by a continuous bijection. This leads us to prove closure properties for the pointclasses Σ ξ 0 in arbitrary recursively presented Polish spaces. We apply our existence results to get better computations of the topological complexity of some sets of dictionaries considered in [D. Lecomte, Omega-powers and descriptive set theory, Journal of Symbolic Logic 70 (4) (2005) 1210–1232].