On stable and unstable behaviour of certain rotation segments

In this paper, we study non-wandering homeomorphisms of the two-dimen-sional torus homotopic to the identity, whose rotation sets are non-trivial segments from (0, 0) to some totally irrational point (α, β). We show that for any r ⩾ 1, this rotation set only appears for C^r diffeomorphisms satisfying some degenerate conditions. And when such a rotation set does appear, assuming several natural conditions that are generically satisfied in the area-preserving world, we give a clearer description of its rotational behaviour. More precisely, the dynamics admits bounded deviation along the direction −(α, β) in the lift, and the rotation set is locked inside an arbitrarily small cone with respect to small C⁰-perturbations of the dynamics. On the other hand, for any non-wandering homeomorphism f with this kind of rotation set, we also present a perturbation scheme in order for the rotation set to be eaten by the rotation set of some nearby dynamics, in the sense that the later set has non-empty interior and contains the former one. These two flavours interplay and share the common goal of understanding the stability/instability properties of this kind of rotation set.