We use persistence modules and their corresponding barcodes to quantitatively distinguish between different fiberwise star-shaped domains in the cotangent bundle of a fixed manifold. The distance between two fiberwise star-shaped domains is measured by a nonlinear version of the classical Banach–Mazur distance, called symplectic Banach–Mazur distance and denoted by dSBM. The relevant persistence modules come from filtered symplectic homology and are stable with respect to dSBM. Our main focus is on the space of unit codisc bundles of orientable surfaces of positive genus, equipped with Riemannian metrics. We consider some questions about large-scale geometry of this space and in particular we give a construction of a quasi-isometric embedding of (ℝn,|⋅| ∞) into this space for all n ∈ ℕ. On the other hand, in the case of domains in T∗S2, we can show that the corresponding metric space has infinite diameter. Finally, we discuss the existence of closed geodesics whose energies can be controlled.

中文翻译：

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持久性模块、辛巴纳赫-马祖尔距离和黎曼度量

我们使用持久性模块及其相应的条形码来定量区分固定流形的余切束中不同的纤维星形域。两个纤维状星形域之间的距离由经典 Banach-Mazur 距离的非线性版本测量，称为辛 Banach-Mazur 距离，表示为dSBM. 相关的持久性模块来自过滤的辛同源性，并且在以下方面是稳定的dSBM. 我们的主要关注点是配备黎曼度量的正属可定向表面的单位 codisc 束空间。我们考虑了一些关于这个空间的大尺度几何的问题，特别是我们给出了一个准等距嵌入的构造(ℝn,|⋅| ∞)为所有人进入这个空间n ∈ ℕ. 另一方面，在域的情况下吨*小号2，我们可以证明对应的度量空间具有无限的直径。最后，我们讨论了能量可以控制的封闭测地线的存在。