Algebras of smooth functions and holography of traversing flows

Let $X$ be a smooth compact manifold and $v$ a vector field on $X$ which admits a smooth function $f:X\to \mathbf{R}$ such that $df(v)>0$. Let $\mathrm{\partial }X$ be the boundary of $X$. We denote by $C^{\mathrm{\infty }}(X)$ the algebra of smooth functions on $X$ and by $C^{\mathrm{\infty }}(\mathrm{\partial }X)$ the algebra of smooth functions on $\mathrm{\partial }X$. With the help of $(v,f)$, we introduce two subalgebras $\mathcal{A}(v)$ and $\mathcal{B}(f)$ of $C^{\mathrm{\infty }}(\mathrm{\partial }X)$ and prove (under mild hypotheses) that $C^{\mathrm{\infty }}(X)\approx \mathcal{A}(v)\widehat{\otimes }\mathcal{B}(f)$, the topological tensor product. Thus the topological algebras $\mathcal{A}(v)$ and $\mathcal{B}(f)$, \emph{viewed as boundary data}, allow for a reconstruction of $C^{\mathrm{\infty }}(X)$. As a result, $\mathcal{A}(v)$ and $\mathcal{B}(f)$ allow for the recovery of the smooth topological type of the bulk $X$.