On Cʳ -generic twist maps of T²

We consider twist diffeomorphisms of the torus, f:T²→ T², and their vertical rotation intervals, ρ V( f)=[ρ V⁻,ρ V⁺], where f is a lift of f to the vertical annulus or cylinder. We show that Cʳ -generically, for any r≥ 1 , both extremes of the rotation interval are rational and locally constant under C⁰ -perturbations of the map. Moreover, when f is area-preserving, Cʳ -generically, ρ V⁻<ρ V⁺ . Also, for any twist map f , f a lift of f to the cylinder, if ρ V⁻<ρ V⁺=p/q , then there are two possibilities: either fq(• )-(0,p) maps a simple essential loop into the connected component of its complement which is below the loop, or it satisfies the curve intersection property. In the first case, ρ V⁺ ≤ p/q in a C⁰ -neighborhood of $f,$ and in the second case, we show that ρ V⁺( f+(0,t))>p/q for all $t>0$ (that is, the rotation interval is ready to grow). Finally, in the Cʳ -generic case, assuming that ρ V⁻<ρ V⁺=p/q, we present some consequences of the existence of the free loop for fq(• )-(0,p) , related to the description and shape of the attractor–repeller pair that exists in the annulus. The case of a Cʳ -generic transitive twist diffeomorphism (if such a thing exists) is also investigated.