  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.
