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

Abstract

For a traversing vector field $v$ on a compact $(n+1)$-manifold $X$ with boundary, we use closed $v$-invariant differential $n$-forms $\Theta$ to define measures $\mu_\Theta$ on the boundary $\partial X$, such that the $v$-flow generated causality map $C_v: \partial^+X(v) \to \partial^-X(v)$ preserves $\mu_\Theta$. In combination with a $\mu_\Theta$-preserving involution $\tau: \partial X \to \partial X$, which maps $\partial^-X(v)$ to $\partial^+X(v)$, the proto-billiard map $B_{v, \tau} := \tau \circ C_v$ is a measure-preserving transformation of $\partial^+X(v)$. We study the dynamical system $(\partial^+X(v), B_{v, \tau})$. Let $(M, g)$ be a Riemannian $n$-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 apply our treatment of $C_v$ and $(\partial^+X(v), B_{v, \tau})$ to the scattering maps $C_{v^g}$ and billiard maps $B_{v^g, \tau}$, where $\tau$ is generated by the reflection of unit tangent vectors with respect to $\partial M$. As a result, 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 tool is the Lyapunov function $F: SM \to \mathbf R$ for $v^g$ and the metrics $g^\bullet$ on $SM$ that make $dF$ harmonic. For such metrics, we get isoperimetric inequalities $vol_{g^\bullet}(SM) \leq vol_{g^\bullet |}(\partial(SM))$. Assuming ergodicity of $B_{v^g, \tau}$ and non-trapping of $g$, we also get formulas for the average length of free geodesic segments in $M$.