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

TL;DR

This paper studies generic properties of twist maps on the torus, showing that their rotation intervals are typically rational and constant under perturbations, with implications for the structure of attractors and dynamical behavior.

Contribution

It establishes generic rationality and constancy of rotation interval endpoints for $C^r$ twist maps and explores consequences for the dynamics and attractor structures.

Findings

Both ends of the rotation interval are rational and locally constant under $C^0$-perturbations.

For area-preserving maps, the rotation interval endpoints are typically distinct.

Existence of free loops influences the shape of attractor-repeller pairs.

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.