LAUSR

We study circle homeomorphisms extensions over a strictly ergodic homeomorphism. Under a very mild restriction, we show that the fibered rotation number is locally constant on an open and dense subset of all circle homeomorphisms extensions homotopic to the trivial extension. In the complement of this set, we find a dense subset in which every map is conjugate to a direct product. Our result provides a generalisation of Avila–Bochi–Damanik’s result on $${\mathrm{SL}}(2,{\mathbb {R}})$$ SL ( 2 , R ) -cocycles, and Jäger–Wang–Zhou’s result on quasi-periodically forced maps, to a broader setting.