高维的Ruelle不变量与凸性
The Ruelle Invariant And Convexity In Higher Dimensions
摘要：
我们构造了任意维保体积流和辛上圈的Ruelle不变量，并证明了几个性质。在$\mathbb{R}^{2n}$中凸区域$X$边界上的线性Reeb流的特殊情况下，我们证明了Ruelle不变量$\text{Ru}(X)$，收缩周期$c(X)$和体积$\text{vol}{X}$满足\[\text{Ru}(X) \cdot c(X) \le C(n) \cdot \text{vol}{X}\]这里$C(n) > 0$是依赖于$n$的显式常数.作为应用，我们构造了$S^{2n-1}$上的非凸的动态凸切触形式，证明了凸性与动态凸性在每个维上的等价性。
英文摘要：
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.
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