Abstract：

Let $X$ be a compact connected smooth manifold with boundary. The paper deals with contact $1$-forms $\beta$ on $X$, whose Reeb vector fields $v_\b$ admit Lyapunov functions $f$. We prove that any odd-dimensional $X$ admits such a contact form.

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 $v_\b$-flow induced ``shadows" on the boundary $\partial X$.

For a given $\beta$, we study the contact vector fields $u$ on $X$ that preserve it invariant. Integrating $u$, we get a $1$-parameter family of contactomorphisms $\{\Phi^t(u)\}_{t \in \mathbb R}$ which maps Reeb trajectories to Reeb trajectories. This leads to estimates from below of the number of $\Phi^t(u)$-fixed Reeb trajectories.

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.