On $C^r$-generic twist maps of ${\mathrm{T}}^2$

We consider twist diffeomorphisms of the torus, $f:{\mathrm{T}}^2\to {\mathrm{T}}^2,$ and their vertical rotation intervals ${\rho }_V(\widehat{f})=[{\rho }_V^{-},{\rho }_V^{+}],$ where $\widehat{f}$ is a lift of $f$ to the vertical annulus or cylinder. We show that $C^r$-generically for any $r\ge 1$, both extremes of the rotation interval are rational and locally constant under $C^0$-perturbations of the map. Moreover, when $f$ is area-preserving, $C^r$-generically ${\rho }_V^{-}<{\rho }_V^{+}\mathrm{.}$ Also, for any twist map $f$, $\widehat{f}$ a lift of $f$ to the cylinder, if ${\rho }_V^{-}<{\rho }_V^{+}=p\mathrm{/}q$, then there are two possibilities: either ${\widehat{f}}^q(\bullet )-(0,p)$ 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, ${\rho }_V^{+}\le p\mathrm{/}q$ in a $C^0$-neighborhood of $f,$ and in the second case, we show that ${\rho }_V^{+}(\widehat{f}+(0,t))>p\mathrm{/}q$ for all $t>0$ (that is, the rotation interval is ready to grow). Finally, in the $C^r$-generic case, assuming that ${\rho }_V^{-}<{\rho }_V^{+}=p\mathrm{/}q,$ we present some consequences of the existence of the free loop for ${\widehat{f}}^q(\bullet )-(0,p)$, related to the description and shape of the attractor-reppeler pair that exists in the annulus. The case of a $C^r$-generic transitive twist diffeomorphism (if such a thing exists) is also investigated.