Let be a diffeomorphism of the closed annulus that preserves orientation and the boundary components, and be a lift of to its universal covering space. Assume that is a Birkhoff region of instability for , and the rotation set of is a nondegenerate interval. Then there exists an open -invariant essential annulus whose frontier intersects both boundary components of , and points and in , such that the positive (resp., negative) orbit of converges to a set contained in the upper (resp., lower) boundary component of and the positive (resp., negative) orbit of converges to a set contained in the lower (resp., upper) boundary component of . This extends a celebrated result originally proved by Mather in the context of area-preserving twist diffeomorphisms.

中文翻译：

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环微分同胚的马瑟不稳定区域

让成为一个闭合环面的微分同胚保留方向和边界分量，并且成为到它的通用覆盖空间。假使，假设是伯克霍夫不稳定区域，以及旋转集是一个非简并区间。那么存在一个开- 不变的本质环其边界与两个边界分量相交，和点和在，使得正（或负）轨道收敛到包含在上（或下）边界分量中的集合和正（分别，负）轨道收敛到包含在下（或上）边界分量中的集合。这扩展了马瑟最初在保面积扭曲微分同胚的背景下证明的著名结果。