Typical Transitivity for Lifts of Rotationless Annulus or Torus Homeomorphisms
Typical Transitivity for Lifts of Rotationless Annulus or Torus Homeomorphisms
复制标题
无旋转环或环面同胚升力的典型传递性
DOI：
10.1112/blms/27.1.79
复制
发表时间：
1995
影响因子：
0.9
通讯作者：
V. Prasad
中科院分区：
文献类型：
--
作者：
S. Alpern;V. Prasad
我们说，如果一个碱基spaceX（可能是环面或环面，n大于或等于2）的同态是无旋转的，如果它是保持面积的，并且对覆盖的spaceX（[0,1] ×RorRn）具有平均平移为零的lifth（∫Ω（h ~ (x) -x）dx=0，其中Ω是[0,1]×[0,1]）。我们证明了（定理1）在具有一致拓扑的X的无旋转同胚空间中，由可传递到X ~的同胚组成的子空间包含一个denseGδ子集。这推广了我们先前的结论，即典型的无旋转同胚具有循环提升，该结论仅在基空间为环空时有效。我们的结果也推广了Besicovitch的结果，Besicovitch在1937年展示了平面的第一个传递同胚。在这种情况下，我们建立了这样一个额外的空间周期的同胚。
We say that a homeomorphismhof the base spaceX(which may be either the annulus orn‐torus,n⩾2) is rotationless if it is area‐preserving and has a lifth∼to the covering spaceX∼([0,1] ×RorRn) with mean translation zero (∫Ω(h∼(x)–x)dx=0, where Ω is [0,1] × [0,1]). We prove (Theorem 1) that in the space of rotationless homeomorphisms of X with the uniform topology, the subspace consisting of homeo‐morphisms with transitive lifts to X ∼ contains a denseGδ subset. This extends our earlier result, valid only when the base space is the annulus, that typical rotationless homeomorphisms have recurrent lifts. Our result also extends that of Besicovitch, who in 1937 exhibited the first transitive homeomorphism of the plane. In this context we establish such a homeomorphism which is additionally spatially periodic.