Holography of Geodesic Flows, Harmonizing Metrics, and Billiards' Dynamics (original) (raw)

Let (M, g) be a Riemannian manifold with boundary, where g is a nontrapping 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 Cvg : ∂ + 1 SM → ∂ − 1 SM , generated by the v g-flow, and the dynamics of the billiard maps Bvg,τ : ∂ + 1 SM → ∂ + 1 SM , where τ denotes an involution, mimicking the elastic reflection from the the boundary ∂M. We getting a variety of holography theorems that tackle the inverse scattering problems for Cvg and theorems that describe the dynamics of Bvg,τ. Our main tools are a Lyapunov function F : SM → R for v g and a special harmonizing Riemannian metrics g • on SM , a metric in which dF is harmonic. For such metrics g • , we get a family of isoperimetric inequalities of the type vol g • (SM) ≤ vol g • | (∂(SM)) and formulas for the average volume of the minimal hypesufaces {F −1 (c)} c∈F (SM). We investigate the interplay between the harmonizing metrics g • and the classical Sasaki metric gg on SM. Assuming ergodicity of Bvg,τ , we also get Santaló-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.