Classical and Effective Descriptive Complexities of ω-Powers

We prove that, for each countable ordinal ξ ≥ 1, there exist some Σ0ξ-complete ω-powers, and some Π0ξ-complete ω-powers, extending previous works on the topological complexity of ω-powers [Fin01, Fin03, Fin04, Lec01, Lec05, DF06]. We prove effective versions of these results; in particular, for each recursive ordinal ξ < ω 1 there exist some recursive sets A ⊆ 2 such that A∞ ∈ Π 0ξ \Σ 0 ξ (respectively,A∈Σ ξ \Π 0 ξ), where Π 0 ξ and Σ 0 ξ denote classes of the hyperarithmetical hierarchy. To do this, we prove effective versions of a result by Kuratowski, describing a Π0ξ set as the range of a closed subset of the Baire space ω by a continuous bijection. This leads us to prove closure properties for the pointclasses Σ ξ 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 [Lec05].