Dynamics of homeomorphisms of the torus homotopic to Dehn twists
Dynamics of homeomorphisms of the torus homotopic to Dehn twists
复制标题
Dehn 扭曲环面同伦同胚动力学
DOI：
10.1017/etds.2012.156
复制
发表时间：
2011
影响因子：
0.9
通讯作者：
B. Garcia
中科院分区：
文献类型：
--
作者：
S. Addas;F. Tal;B. Garcia
摘要本文考虑与Dehn扭同伦的环面同胚。我们证明，如果$f$的垂直旋转集减少到零，那么存在一个紧凑的连接本质的“水平”集$K$，不变下$f$。换句话说，如果我们考虑$f$的升力$\hat {f}$到垂直旋转数为零的圆柱，则所有点在$\hat {f}$的迭代下都有一致有界运动。此外，我们给出了一个简单的明确的条件，当满足时，意味着垂直旋转集包含一个区间，从而也意味着积极的拓扑熵。作为上述结果的推论，我们证明了Boyland猜想的一个版本：如果$f$是面积保持的，并且具有到具有零Lebesgue测度垂直旋转数的圆柱的提升$\hat {f}$，则所有点的轨道在$\hat {f}$下一致有界，或者圆柱中有正垂直速度的点和负垂直速度的点。
Abstract In this paper, we consider torus homeomorphisms $f$ homotopic to Dehn twists. We prove that if the vertical rotation set of $f$ is reduced to zero, then there exists a compact connected essential ‘horizontal’ set $K$, invariant under $f$. In other words, if we consider the lift $\hat {f}$ of $f$ to the cylinder, which has zero vertical rotation number, then all points have uniformly bounded motion under iterates of $\hat {f}$. Also, we give a simple explicit condition which, when satisfied, implies that the vertical rotation set contains an interval and thus also implies positive topological entropy. As a corollary of the above results, we prove a version of Boyland’s conjecture to this setting: if $f$ is area preserving and has a lift $\hat {f}$ to the cylinder with zero Lebesgue measure vertical rotation number, then either the orbits of all points are uniformly bounded under $\hat {f}$, or there are points in the cylinder with positive vertical velocity and others with negative vertical velocity.