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 $C_{v^g}:{\mathrm{\partial }}_1^{+}SM\to {\mathrm{\partial }}_1^{-}SM$, 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 $\mathrm{\partial }M$. 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 $vo{l}_{g^{\bullet }}(SM)\le vo{l}_{g^{\bullet }\mathrm{\mid }}(\mathrm{\partial }(SM))$ and formulas for the average volume of the minimal hypesufaces ${F^{-1}©}{c \in F(SM)}$\mathrm{.}Weinvestigatetheinterplaybetweentheharmonizingmetrics$g^\bullet$andtheclassicalSasakimetric$gg$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.