Mathers regions of instability for annulus diffeomorphisms

Let $f$ be a $C^{1+\varepsilon}$ diffeomorphism of the closed annulus $A$ that preserves orientation and the boundary components, and $\widetilde{f}$ be a lift of $f$ to its universal covering space. Assume that $A$ is a Birkhoff region of instability for $f$, and the rotation set of $\widetilde{f}$ is a non-degenerate interval. Then there exists an open $f$-invariant annulus $A^*$ whose boundary intersects both boundary components of of $A$, and points $z^+$ and $z^-$ in $A^*$, such that the positive (resp. negative) orbit of $z^+$ converges to a set contained in the upper (resp. lower) boundary component of $A^*$ and the positive (resp. negative) orbit of $z^-$ converges to a set contained in the lower (resp. upper) boundary component of $A^*$. This extends a celebrated result originally proved by Mather for area-preserving twist diffeomorphisms.