The Ruelle invariant and convexity in higher dimensions

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 \operatorname{Ru}(X) , the period of the systole c(X) and the volume \operatorname{vol}(X) satisfy \operatorname{Ru}(X) \cdot c(X) \le C(n) \cdot \operatorname{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.