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arXiv · 2205.00935

The Ruelle Invariant And Convexity In Higher Dimensions

Julian Chaidez·Oliver Edtmair
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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.

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Julian Chaidez, Oliver Edtmair. 2022-05-02. The Ruelle Invariant And Convexity In Higher Dimensions. https://arxiv.org/abs/2205.00935

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