On topological genericity of the mode-locking phenomenon

TL;DR

This paper investigates the topological properties of mode-locking in circle homeomorphisms extended over ergodic systems, showing local constancy of rotation numbers and conjugacy to product maps in dense subsets, generalizing previous results.

Contribution

It extends known results on mode-locking phenomena to a broader setting involving circle homeomorphisms over ergodic systems, highlighting topological genericity.

Findings

Fibered rotation number is locally constant on an open dense set.

Dense subset where maps are conjugate to a direct product.

Generalizes results from SL(2,R)-cocycles and quasi-periodic maps.

Abstract

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 ${\rm SL}(2,\mathbb{R})-$cocycles, and Jager-Wang-Zhou's result on quasi-periodically forced maps, to a broader setting.