论文标题
测量流，协调指标和台球动态的全息图
Holography of geodesic flows, harmonizing metrics, and billiards' dynamics
论文作者
论文摘要
令$（m，g）$为带边界的Riemannian歧管，其中$ g $是非捕获度量的。让$ sm $是球形切线与$ m $捆绑的空间，而$ v^g $ the Geodesic vector Field在$ sm $上。我们研究散射映射$ c_ {v^g}：\ partial^+_ 1sm \ to \ partial^-_ 1sm $，由$ v^g $ -Flow生成，台球的动态图$ b_ {v^g，τ}相互作用，模仿来自边界$ \ partial m $的弹性反射。我们获得了各种全息定理，这些定理解决了$ c_ {v^g} $的反向散射问题和描述$ b_ {v^g，τ} $的动力学的定理。我们的主要工具是lyapunov函数$ f：sm \ to \ mathbb r $ for $ v^g $和一个特殊的协调riemannian指标$ g^\ bullet $ on $ sm $，这是$ df $谐波的度量。对于此类指标$ g^\ bullet $，我们得到了$ vol_ {g^\ bullet}型类型的等等不平等，\ leq vol_ {g^\ bullet |}（\ bulter |}（\ partial（sm）$和最小值$ \ \ \ \ f^f^f^{c {c）的平均水平和公式f（sm）} $。我们研究了$ sm $上的统一指标$ g^\ bullet $与经典的sasaki公制$ gg $之间的相互作用。假设$ b_ {v^g，τ} $的牙齿性，我们还获得了$ m $的免费地球段的平均长度，以及沿$ v^g $ -trajextories的平均变化的平均长度和lyapunov函数$ f $的平均变化的平均长度。
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 $B_{v^g, τ}: \partial^+_1SM \to \partial^+_1SM$, where $τ$ 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 $B_{v^g, τ}$. 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 $B_{v^g, τ}$, 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.
