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Let $X subset mathbb{R}^4$ be a convex domain with smooth boundary $Y$. We use a relation between the extrinsic curvature of $Y$ and the Ruelle invariant $text{Ru}(Y)$ of the natural Reeb flow on $Y$ to prove that there exist constants $C > c > 0$ independent of $Y$ such that [c < frac{text{Ru}(Y)^2}{text{vol}(X)} cdot text{sys}(Y) < C] Here $text{sys}(Y)$ is the systolic ratio, i.e. the square of the minimal period of a closed Reeb orbit of $Y$ divided by twice the volume of $X$. We then construct dynamically convex contact forms on $S^3$ that violate this bound using methods of Abbondandolo-Bramham-Hryniewicz-Salom~{a}o. These are the first examples of dynamically convex contact $3$-spheres that are not strictly contactomorphic to a convex boundary $Y$.
We prove that Besse contact forms on closed connected 3-manifolds, that is, contact forms with a periodic Reeb flow, are the local maximizers of suitable higher systolic ratios. Our result extends earlier ones for Zoll contact forms, that is, contact
We establish an equidistribution result for Ruelle resonant states on compact locally symmetric spaces of rank one. More precisely, we prove that among the first band Ruelle resonances there is a density one subsequence such that the respective produ
We prove exponential decay of correlations for Holder continuous observables with respect to any Gibbs measure for contact Anosov flows admitting Pesin sets with exponentially small tails. This is achieved by establishing strong spectral estimates fo
A differential 1-form $alpha$ on a manifold of odd dimension $2n+1$, which satisfies the contact condition $alpha wedge (dalpha)^n eq 0$ almost everywhere, but which vanishes at a point $O$, i.e. $alpha (O) = 0$, is called a textit{singular contact
We consider convex contact spheres $Y$ all of whose Reeb orbits are closed. Any such $Y$ admits a stratification by the periods of closed Reeb orbits. We show that $Y$ resembles a contact ellipsoid: any stratum of $Y$ is an integral homology sphere,