Holography of geodesic flows, harmonizing metrics, and billiards' dynamics

Let $(M, g)$ be a Riemannian manifold with boundary, where $g$ is a non-trapping metric. Let $SM$ be the space of the spherical tangent to $M$ bundle, and $v^g$ the geodesic vector field on $SM$. We study the scattering maps $C_{v^g}: \partial^+_1SM \to \partial^-_1SM$, generated by the $v^g$-flow, and the dynamics of the billiard maps $B_{v^g, \tau}: \partial^+_1SM \to \partial^+_1SM$, where $\tau$ denotes an involution, mimicking the elastic reflection from the the boundary $\partial M$. We getting a variety of holography theorems that tackle the inverse scattering problems for $C_{v^g}$ and theorems that describe the dynamics of $B_{v^g, \tau}$. Our main tools are a Lyapunov function $F: SM \to \mathbb R$ for $v^g$ and a special harmonizing Riemannian metrics $g^\bullet$ on $SM$, a metric in which $dF$ is harmonic. For such metrics $g^\bullet$, we get a family of isoperimetric inequalities of the type $vol_{g^\bullet}(SM) \leq vol_{g^\bullet |}(\partial(SM))$ and formulas for the average volume of the minimal hypesufaces $\{F^{-1}(c)\}_{c \in F(SM)}$. We investigate the interplay between the harmonizing metrics $g^\bullet$ and the classical Sasaki metric $gg$ on $SM$. Assuming ergodicity of $B_{v^g, \tau}$, we also get Santal\'{o}-Chernov type formulas for the average length of free geodesic segments in $M$ and for the average variation of the Lyapunov function $F$ along the $v^g$-trajectories.