论文标题
较高维度的不变和凸状
The Ruelle Invariant And Convexity In Higher Dimensions
论文作者
论文摘要
我们在任何维度上构建了保持体积的流量和符号共生的不变性，并证明了几种属性。在$ \ mathbb {r}^{2n} $的凸域$ x $边界上的线性reeb流的特殊情况下，我们证明了ruelle不变$ \ text {ru} {ru}（x）$ \ [\ text {ru}（x）\ cdot c（x）\ le c（n）\ cdot \ text {vol} {x} {x} \]这里$ c（n）> 0 $是$ n $的明确常数。作为一个应用程序，我们在$ s^{2n-1} $上动态构建凸触点表单，而不是凸的，在每个维度中都表达了凸度和动态凸度的等效性。
We construct the Ruelle invariant of a volume preserving flow and a symplectic cocycle in any dimension and prove several properties. In the special case of the linearized Reeb flow on the boundary of a convex domain $X$ in $\mathbb{R}^{2n}$, we prove that the Ruelle invariant $\text{Ru}(X)$, the period of the systole $c(X)$ and the volume $\text{vol}{X}$ satisfy \[\text{Ru}(X) \cdot c(X) \le C(n) \cdot \text{vol}{X}\] Here $C(n) > 0$ is an explicit constant dependent on $n$. As an application, we construct dynamically convex contact forms on $S^{2n-1}$ that are not convex, disproving the equivalence of convexity and dynamical convexity in every dimension.
