Recovering contact forms from boundary data

Abstract

Let $X$ be a compact smooth manifold with boundary. The paper deals with contact $1$-forms $\beta$ on $X$, whose Reeb vector fields $v_\beta$ admit Lyapunov functions $f$. We tackle the question: how to recover $X$ and $\beta$ from the appropriate data along the boundary $\partial X$? We describe such boundary data and prove that they allow for a reconstruction of the pair $(X, \beta)$, up to a diffeomorphism of $X$. We use the term ``holography" for the reconstruction. We say that objects or structures inside $X$ are {\it holographic}, if they can be reconstructed from their $v_\beta$-flow induced ``shadows" on the boundary $\partial X$. We also introduce numerical invariants that measure how ``wrinkled" the boundary $\partial X$ is with respect to the $v_\beta$-flow and study their holographic properties under the contact forms preserving embeddings of equidimensional contact manifolds with boundary. We get some ``non-squeezing results" about such contact embedding, which are reminiscent of Gromov's non-squeezing theorem in symplectic geometry.