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

Abstract

We consider twist diffeomorphisms of the torus, $f:{\rm T^2\rightarrow T^2,}$ 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 $r\geq 1$, 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 $\rho _V^{-}<\rho _V^{+}.$ Also, for any twist map $f$, $\widehat{f}$ a lift of $f$ to the cylinder, if $\rho _V^{-}<\rho _V^{+}=p/q$, 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, $\rho _V^{+} \leq p/q$ in a $C^0$-neighborhood of $f,$ and in the second case, we show that $\rho _V^{+}(\widehat{f}+(0,t))>p/q$ for all $t>0$ (that is, the rotation interval is ready to grow). Finally, in the $C^r$-generic case, assuming that $\rho _V^{-}<\rho _V^{+}=p/q,$ 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.