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 R²ⁿ , we prove that the Ruelle invariant Ru(X) , the period of the systole c(X) and the volume vol(X) satisfy Ru(X) · c(X) ≤ C(n) · vol(X) . Here C(n) > 0 is an explicit constant depending on n . As an application, we construct dynamically convex contact forms on S²ⁿ⁻¹ that are not convex, disproving the equivalence of convexity and dynamical convexity in every dimension.