The Ruelle Invariant And Convexity In Higher Dimensions

Abstract

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.

29 pages, 2 figures. Comments welcome!

Cited by in corpus (1)

• Coarse distance from dynamically convex to convex