Recovering contact forms from boundary data

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 $\mathrm{\partial }X$? 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 $v_β$-flow induced shadows" on the boundary $\mathrm{\partial }X$. We also introduce numerical invariants that measure how wrinkled" the boundary $\partial X$ is with respect to the $v_β$-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.