标题： 关于安诺索夫流空间的类似霍费尔度量 显示英文标题

标题： A Hofer-like Metric on the Space of Anosov Flows

摘要： 本文发展了一组类似于Hofer度量的度量($\dAnV{V}$)，用于Anosov向量场空间($\An(M)$)，通过使用($\Ck{k}$)或Sobolev($\SobolevHk{k}$)范数来提供基于变形路径代价的动力学相关距离。 我们建立了基本性质，包括对于($V= C^r (r \ge 1)$)或($H^k (k > \dim(M)/2+1)$)的完备性，以及在微分同胚下的自然性。 我们通过证明定量稳定性结果展示了这些度量的实用性：在($\dAnV{V}$)中的接近性意味着核心动力学不变量的变化受到控制，包括拓扑熵、李雅普诺夫指数、SRB测度、热力学压力、谱隙（混合速率）和ζ函数。 足够的正则性确保了局部Lipschitz连续性和Fréchet可微性，将度量结构与线性响应公式联系起来，特别是对于压力、指数和谱隙。 虽然Sobolev度量产生了局部平坦的几何形状和直线测地线，但该框架具有广泛适用性。 我们探讨了它对Anosov流模空间的影响，包括不变量的稳定性以及局部切片定理的框架。 此外，我们引入了\emph{拓扑Anosov流}，它通过生成场的同时均匀流收敛和$\dAn$度量收敛来定义。 这一新类旨在捕捉非光滑设置中的基本双曲特征。 总体而言，所提出的度量提供了多个几何视角，用于分析 Anosov 动力学的稳定性、分类及其可能的扩展。 显示英文摘要

摘要： This paper develops a family of Hofer-like metrics ($\dAnV{V}$) on the space of Anosov vector fields $\An(M)$, providing dynamically relevant distances based on the cost of deformation paths using $\Ck{k}$ or Sobolev $\SobolevHk{k}$ norms. We establish fundamental properties, including completeness for $V= C^r (r \ge 1)$ or $H^k (k > \dim(M)/2+1)$, and naturality under diffeomorphisms. We show the utility of these metrics by proving quantitative stability results: proximity in $\dAnV{V}$ implies controlled variation of essential dynamical invariants, including topological entropy, Lyapunov exponents, SRB measures, thermodynamic pressure, spectral gaps (mixing rates), and zeta functions. Sufficient regularity ensures local Lipschitz continuity and Fr\'echet differentiability, connecting the metric structure to linear response formulas, particularly for pressure, exponents, and the spectral gap. While Sobolev metrics yield locally flat geometry with straight line geodesics, the framework is broadly applicable. We explore implications for the moduli space of Anosov flows, including stability of invariants and the framework for local slice theorems. Furthermore, we introduce \emph{Topological Anosov Flows}, defined via simultaneous uniform flow convergence and $\dAn$ metric convergence of the generating fields. This new class aims to capture essential hyperbolic features in non-smooth settings. Overall, the proposed metrics offer several geometric perspectives for analyzing the stability, classification, and possible extensions of Anosov dynamics.