Wadge degrees of $\Delta^0_2$ omega-powers

TL;DR

This paper constructs specific regular languages whose omega-powers are complete for various Wadge degrees within certain classes of Borel sets, advancing the understanding of the descriptive complexity of omega-powers.

Contribution

It provides explicit regular languages with omega-powers that are complete for each class among several levels of the Borel hierarchy, detailing their Wadge degrees.

Findings

Constructed regular languages with omega-powers complete for each class.

Established the Wadge degrees of these omega-powers within the hierarchy.

Enhanced the classification of omega-powers in descriptive set theory.

Abstract

We provide, for each natural number $n$ and each class among $D_n(\Sigma^0_1)$, $\bar D_n(\Sigma^0_1)$ and $D_{2n+1}(\Sigma^0_1)\oplus\bar D_{2n+1}(\Sigma^0_1)$, a regular language whose associated omega-power is complete for this class.