A Hofer-like Metric on the Space of Anosov Flows

Abstract

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.