The Ruelle Invariant And Convexity In Higher Dimensions

2022/05/02 by Julian Chaidez, Chaidez, Julian, Oliver Edtmair +1 · 2 citations
Mathematics · #Mathematical Dynamics and Fractals #Geometry and complex manifolds #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2205.00935

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 ℝ²ⁿ, we prove that the Ruelle invariant Ru(X), the period of the systole c(X) and the volume volX satisfy Ru(X) ⋅ c(X) ≤ C(n) ⋅ volX 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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• Topological Obstructions to Dynamical Convexity
• On dynamical invariants of coadjoint orbits of 3D volume-preserving diffeomorphisms

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