用 Lyapunov 谱刻画对称空间
Characterizing symmetric spaces by their Lyapunov spectra
AI总结：
本文通过构造水平测度及其 Lyapunov 谱和水平维数，证明闭负曲率局部对称空间可由测地流周期轨道的 Lyapunov 谱在等距意义下唯一刻画，并推广到光滑 Anosov 流的轨道等价刚性。
摘要：
我们证明，闭负曲率局部对称空间在所有同伦等价的负曲率流形中，可以通过其测地流周期轨道的 Lyapunov 谱在等距意义下被刻画。这是通过构造一种新的测地流不变测度来实现的，我们称之为水平测度。我们证明，仅凭水平测度的 Lyapunov 谱就足以在局部上将这些局部对称空间刻画到等距。我们将一个新的不变量，即水平维数，与水平测度联系起来。我们将这一不变量与复双曲流形的曲率夹挤刚性定理的推广联系起来，从而得到 Lyapunov 谱的夹挤刚性定理。我们的方法可以推广，给出光滑 Anosov 流 $f^{t}$ 的刚性定理，其中 $f^{t}$ 与闭负曲率局部对称空间 $X$ 的测地流 $g^{t}_{X}$ 轨道等价：$f^{t}$ 与 $g^{t}_{X}$ 光滑轨道等价当且仅当它在所有周期轨道上的 Lyapunov 谱是 $g^{t}_{X}$ 相应 Lyapunov 谱的倍数。
英文摘要：
We prove that closed negatively curved locally symmetric spaces are characterized up to isometry among all homotopy equivalent negatively curved manifolds by the Lyapunov spectra of the periodic orbits of their geodesic flows. This is done by constructing a new invariant measure for the geodesic flow that we refer to as the horizontal measure. We show that the Lyapunov spectrum of the horizontal measure alone suffices to locally characterize these locally symmetric spaces up to isometry. We associate to the horizontal measure a new invariant, the horizontal dimension. We tie this invariant to extensions of curvature pinching rigidity theorems for complex hyperbolic manifolds to pinching rigidity theorems for the Lyapunov spectrum. Our methods extend to give a rigidity theorem for smooth Anosov flows $f^{t}$ orbit equivalent to the geodesic flow $g^{t}_{X}$ of a closed negatively curved locally symmetric space $X$: $f^{t}$ is smoothly orbit equivalent to $g^{t}_{X}$ if and only if its Lyapunov spectra on all periodic orbits are a multiple of the corresponding Lyapunov spectra for $g^{t}_{X}$.
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