论文标题
Anosov以$ 3 $ -Manifolds流动：手术和叶子
Anosov flows on $3$-manifolds: the surgeries and the foliations
论文作者
论文摘要
在3个manifold Fe1上的任何Anosov流X上，都将双旋转平面（一个赋予两个横向叶子FS和FU的平面），该平面反映了赋予中心稳定和中心不稳定叶子的流动的正常结构。如果FS（或FU）微不足道，则将流量覆盖。 从一个Anosov流动可以通过Dehn-Goodman Fried手术进行无限的其他许多人。本文研究了这些手术如何修饰双分离平面。 我们首先注意到，沿某些特定周期轨道的手术不会修改双层平面：例如， - 对应于对应于不相关的简单闭合大地测量的轨道家族的手术不会影响与双曲线表面的大地测量流相关的双溶液平面（定理1）； -FE2将一个（非空的）有限的周期性轨道家族（称为枢轴）与任何非R覆盖的Anosov流相关联。枢轴上的手术不会影响双分离平面的分支结构（定理2） 我们考虑了Dehn-Goodman Fried Garregery从Anosov Automorimists a的悬架流中获得的Anosov流动的固定外科手术（a）。 每项此类手术都与有限的夫妻（C，M（C））相关，其中C是周期性的轨道和M（C）整数。当所有M（C）具有相同的符号时，Fenley证明了诱导的Anosov流由R覆盖并根据手术符号扭曲。我们在这里分析了手术在有限的X和另一组Y上为负面的阳性的情况。 除其他结果外，我们还表明，鉴于外科手术中的任何流X： - 存在E> 0，因此，对于每个e密集的周期性轨道C，通过沿C的非琐事手术从X获得的每个流量都被R覆盖（定理4）。 - 存在周期性轨道c+，c-，使得通过x从x获得的每个流量在C+和C-上具有不同符号的手术均为非R覆盖（定理5）。
To any Anosov flow X on a 3-manifold Fe1 associated a bi-foliated plane (a plane endowed with two transverse foliations Fs and Fu) which reflects the normal structure of the flow endowed with the center-stable and center unstable foliations. A flow is R-covered if Fs (or Fu) is trivial. From one Anosov flow one can build infinitely many others by Dehn-Goodman-Fried surgeries. This paper investigates how these surgeries modify the bi-foliated plane. We first noticed that surgeries along some specific periodic orbits do not modify the bi-foliated plane: for instance, - surgeries on families of orbits corresponding to disjoint simple closed geodesics do not affect the bi-foliated plane associated to the geodesic flow of a hyperbolic surface (Theorem 1); - Fe2 associates a (non-empty) finite family of periodic orbits, called pivots, to any non-R-covered Anosov flow. Surgeries on pivots do not affect the branching structure of the bi-foliated plane (Theorem 2) We consider the set Surg(A) of Anosov flows obtained by Dehn-Goodman-Fried surgery from the suspension flows of Anosov automorphisms A in SL(2,Z) of the torus T2. Every such surgery is associated to a finite set of couples (C,m(C)), where the C are periodic orbits and the m(C) integers. When all the m(C) have the same sign, Fenley proved that the induced Anosov flow is R-covered and twisted according to the sign of the surgery. We analyse here the case where the surgeries are positive on a finite set X and negative on another set Y. Among other results, we show that given any flow X in Surg(A) : - there exists e>0 such that for every e-dense periodic orbit C, every flow obtained from X by a non trivial surgery along C is R-covered (Theorem 4). - there exist periodic orbits C+,C- such that every flow obtained from X by surgeries with distinct signs on C+ and C- is non-R-covered (Theorem 5).
