On topological genericity of the mode-locking phenomenon

We study the 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. In the complement of this set, we found 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 $\mathrm{S}\mathrm{L}(2,\mathbb{R})-$cocycles, and Jager-Wang-Zhou’s result on quasi-periodically forced maps, to a broader setting.