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Resonant forms at zero for dissipative Anosov flows

Mihajlo Cekić, Gabriel P. Paternain

TL;DR

This work develops a comprehensive microlocal framework for resonances at zero of dissipative Anosov flows on 3-manifolds, centered on SRB measures and their associated winding cycles. It introduces helicity as a robust invariant when both winding cycles vanish, and proves horocyclic invariance and perturbative structure, including a Banach-manifold picture for flows with vanishing winding data. The paper provides explicit analyses for quasi-Fuchsian flows and thermostats, derives formulas for the derivative of winding cycles, establishes (in many cases) semisimplicity of the zero-resonance action, and connects helicity to a linking form of SRB measures and closed orbits. Altogether, it offers a unifying approach to resonances, dynamical invariants, and zeta-function behavior in dissipative 3D-Anosov dynamics, with concrete computations in notable models and broad implications for stability under time changes and perturbations.

Abstract

We study resonant differential forms at zero for transitive Anosov flows on $3$-manifolds. We pay particular attention to the dissipative case, that is, Anosov flows that do not preserve an absolutely continuous measure. Such flows have two distinguished Sinai-Ruelle-Bowen $3$-forms, $Ω_{\text{SRB}}^{\pm}$, and the cohomology classes $[ι_{X}Ω_{\text{SRB}}^{\pm}]$ (where $X$ is the infinitesimal generator of the flow) play a key role in the determination of the space of resonant $1$-forms. When both classes vanish we associate to the flow a $\textit{helicity}$ that naturally extends the classical notion associated with null-homologous volume preserving flows. We provide a general theory that includes horocyclic invariance of resonant $1$-forms and SRB-measures as well as the local geometry of the maps $X\mapsto [ι_{X}Ω_{\text{SRB}}^{\pm}]$ near a null-homologous volume preserving flow. Next, we study several relevant classes of examples. Among these are thermostats associated with holomorphic quadratic differentials, giving rise to quasi-Fuchsian flows as introduced by Ghys. For these flows we compute explicitly all resonant $1$-forms at zero, we show that $[ι_{X}Ω_{\text{SRB}}^{\pm}]=0$ and give an explicit formula for the helicity. In addition we show that a generic time change of a quasi-Fuchsian flow is semisimple and thus the order of vanishing of the Ruelle zeta function at zero is $-χ(M)$, the same as in the geodesic flow case. In contrast, we show that if $(M,g)$ is a closed surface of negative curvature, the Gaussian thermostat driven by a (small) harmonic $1$-form has a Ruelle zeta function whose order of vanishing at zero is $-χ(M)-1$.

Resonant forms at zero for dissipative Anosov flows

TL;DR

This work develops a comprehensive microlocal framework for resonances at zero of dissipative Anosov flows on 3-manifolds, centered on SRB measures and their associated winding cycles. It introduces helicity as a robust invariant when both winding cycles vanish, and proves horocyclic invariance and perturbative structure, including a Banach-manifold picture for flows with vanishing winding data. The paper provides explicit analyses for quasi-Fuchsian flows and thermostats, derives formulas for the derivative of winding cycles, establishes (in many cases) semisimplicity of the zero-resonance action, and connects helicity to a linking form of SRB measures and closed orbits. Altogether, it offers a unifying approach to resonances, dynamical invariants, and zeta-function behavior in dissipative 3D-Anosov dynamics, with concrete computations in notable models and broad implications for stability under time changes and perturbations.

Abstract

We study resonant differential forms at zero for transitive Anosov flows on -manifolds. We pay particular attention to the dissipative case, that is, Anosov flows that do not preserve an absolutely continuous measure. Such flows have two distinguished Sinai-Ruelle-Bowen -forms, , and the cohomology classes (where is the infinitesimal generator of the flow) play a key role in the determination of the space of resonant -forms. When both classes vanish we associate to the flow a that naturally extends the classical notion associated with null-homologous volume preserving flows. We provide a general theory that includes horocyclic invariance of resonant -forms and SRB-measures as well as the local geometry of the maps near a null-homologous volume preserving flow. Next, we study several relevant classes of examples. Among these are thermostats associated with holomorphic quadratic differentials, giving rise to quasi-Fuchsian flows as introduced by Ghys. For these flows we compute explicitly all resonant -forms at zero, we show that and give an explicit formula for the helicity. In addition we show that a generic time change of a quasi-Fuchsian flow is semisimple and thus the order of vanishing of the Ruelle zeta function at zero is , the same as in the geodesic flow case. In contrast, we show that if is a closed surface of negative curvature, the Gaussian thermostat driven by a (small) harmonic -form has a Ruelle zeta function whose order of vanishing at zero is .
Paper Structure (50 sections, 64 theorems, 354 equations, 1 figure)

This paper contains 50 sections, 64 theorems, 354 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Geometric multiplicities of resonant forms
  3. Helicity of the SRB measures
  4. Perturbation theory
  5. Horocyclic invariance of resonant forms
  6. Semisimplicity
  7. Examples
  8. Suspensions
  9. Quasi-Fuchsian flows and the coupled vortex equations
  10. Flows with $[\omega^{+}]=0$ and $[\omega^{-}]\neq 0$.
  11. Homologically full flows with $[\omega^{\pm}]\neq 0$
  12. Outline of the paper
  13. Preliminaries
  14. The resolvent and Pollicott-Ruelle resonances
  15. The pairings
  16. ...and 35 more sections

Key Result

Theorem 1.1

Assume $X$ generates a transitive Anosov flow on $\mathcal{M}$. The geometric multiplicity for the action of $\mathop{\mathrm{\mathcal{L}}}\nolimits_X$ on $\Omega_0^1$ and $\Omega^1$ is determined as a function of $[\omega^\pm]_{H^2(\mathcal{M})}$ and $\mathcal{H}(X)$ by the following table: Moreover, the map $\mathop{\mathrm{Res}}\nolimits^1\cap \ker d\ni u\mapsto [u]\in H^{1}(\mathcal{M})$ is

Figures (1)

  • Figure 1: Local geometry of vector fields according to the value of the winding cycle and the helicity. Flows $X_s \in \mathcal{W}^+$, but $X_s \not \in \mathcal{W}^-$, are constructed in Proposition \ref{['prop:newexamples']}, quasi-Fuchsian flows $X_{\mathrm{qF}} \in \mathcal{W}$ are studied in Section \ref{['sec:qhd']}, and thermostat flows satisfying $X_{\mathrm{th}} \not \in \mathcal{W}^+ \cup \mathcal{W}^-$ are provided by Theorem \ref{['thm:beta']}.

Theorems & Definitions (145)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Corollary 1.4
  • proof
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • ...and 135 more