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 $\mathrm{Ru}(X)$, the period of the systole $c(X)$ and the volume $\mathrm{vol}(X)$ satisfy $\mathrm{Ru}(X)\cdot c(X)\le C(n)\cdot \mathrm{vol}(X)$. Here $C(n)>0$ is an explicit constant depending 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.