Drift orbits for families of twist maps of the annulus

Abstract

We generalize the classical result of J. Mather stating the existence of a drift orbit inside a region of instability of an exact-symplectic positive twist map, to the case of a finite family ${\cal F}$ of such maps. A special case is the case where the maps $F\in{\cal F}$ have no common invariant continuous graph. We prove the existence of a sequence $(z_i)_{i\in\mathbb{Z}}$ in $\mathbb{R}/\mathbb{Z}\times\mathbb{R}$ and of a sequence $(F_i)_{i\in\mathbb{Z}}$ in ${\cal F}$ such that

\[ z_{i+1}=F_i(z_i),\quad \lim_{i\to -\infty} p_2(z_i)=-\infty,\quad \lim_{i\to +\infty} p_2(z_i)=+\infty, \]

where $p_{2} :(x,y)\mapsto y$ is the second projection.