Complete axiomatization of the stutter-invariant fragment of the linear time µ-calculus

The logic µ(U) is the fixpoint extension of the "Until"-only fragment of linear-time temporal logic. It also happens to be the stutter-invariant fragment of linear-time µ-calculus µ(◊). We provide complete axiomatizations of µ(U) on the class of finite words and on the class of ω-words. We introduce for this end another logic, which we call µ(◊_Γ), and which is a variation of µ(◊) where the Next time operator is replaced by the family of its stutter-invariant counterparts. This logic has exactly the same expressive power as µ(U). Using already known results for µ(◊), we first prove completeness for µ(◊_Γ), which finally allows us to obtain completeness for µ(U).