Uniqueness vs non-uniqueness in complete connections with modified majority rules

We take a closer look at a class of chains with complete connections introduced by Berger, Hoffman and Sidoravicius. Besides giving a sharper description of the uniqueness and non-uniqueness regimes, we show that if the pure majority rule used to fix the dependence on the past is replaced with a function that is Lipschitz at the origin, then uniqueness always holds, even with arbitrarily slow decaying variation.