The Ruelle Invariant And Convexity In Higher Dimensions

TL;DR

This paper introduces the Ruelle invariant for volume-preserving flows and symplectic cocycles in any dimension, establishing inequalities relating it to convex domains and demonstrating that dynamical convexity does not imply convexity in higher dimensions.

Contribution

It constructs the Ruelle invariant in higher dimensions, proves key inequalities involving convex domains, and shows dynamical convexity does not imply convexity in all dimensions.

Findings

The Ruelle invariant satisfies specific inequalities with volume and systole in convex domains.

Existence of dynamically convex contact forms that are not convex in higher dimensions.

Disproof of the equivalence between convexity and dynamical convexity in all 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.