[Paper Review] Resonant spaces for volume preserving Anosov flows

Mihajlo Cekić, Gabriel P. Paternain|arXiv (Cornell University)|Nov 10, 2019

Mathematical Dynamics and Fractals39 references4 citations

TL;DR

This paper computes the dimensions of resonant spaces for volume-preserving Anosov flows on 3-manifolds using the first Betti number, the cohomology class $[\iota_X\Omega]$, and helicity. It establishes that these dimensions match Pollicott-Ruelle resonance multiplicities under semisimplicity, proves semisimplicity fails for time-changed hyperbolic geodesic flows, and shows that non-null-homologous deformations preserve semisimplicity, with implications for the zero-order vanishing of the Ruelle zeta function.

ABSTRACT

We consider Anosov flows on closed 3-manifolds preserving a volume form $Ω$. Following Dyatlov and Zworski (2017) we study spaces of invariant distributions with values in the bundle of exterior forms whose wavefront set is contained in the dual of the unstable bundle. Our first result computes the dimension of these spaces in terms of the first Betti number of the manifold, the cohomology class $[ι_{X}Ω]$ (where $X$ is the infinitesimal generator of the flow) and the helicity. These dimensions coincide with the Pollicott-Ruelle resonance multiplicities under the assumption of $ extit{semisimplicity}$. We prove various results regarding semisimplicity on 1-forms, including an example showing that it may fail for time changes of hyperbolic geodesic flows. We also study non null-homologous deformations of contact Anosov flows and we show that there is always a splitting Pollicott-Ruelle resonance on 1-forms and that semisimplicity persists in this instance. These results have consequences for the order of vanishing at zero of the Ruelle zeta function. Finally our analysis also incorporates a flat unitary twist in both, the resonant spaces and the Ruelle zeta function.

Motivation & Objective

• To compute the dimensions of resonant spaces of invariant distributions with wave front set in the dual unstable bundle for volume-preserving Anosov flows on 3-manifolds.
• To relate these dimensions to Pollicott-Ruelle resonance multiplicities under semisimplicity conditions.
• To investigate the validity of 1-semisimplicity for volume-preserving Anosov flows, especially in time-changed hyperbolic geodesic flows.
• To study the behavior of resonant spaces and the Ruelle zeta function under non-null-homologous deformations of contact Anosov flows.
• To analyze the impact of flat unitary twists on resonant spaces and the zeta function.

Proposed method

• Define resonant spaces $\text{Res}_k(0)$ as distributions in $\mathcal{D}'_{E_u^*}(M;\Omega^k)$ annihilated by $\iota_X$ and $\iota_X d$.
• Use the first Betti number $b_1(M)$, the cohomology class $[\iota_X\Omega]$, and the helicity $\mathcal{H}(X)$ to compute $\dim\text{Res}_k(0)$.
• Prove that $\dim\text{Res}_k(0)$ equals the Pollicott-Ruelle resonance multiplicity $m_k(0)$ under semisimplicity.
• Construct an example where 1-semisimplicity fails for time-changed hyperbolic geodesic flows, using Hölder continuity of the stable/unstable bundles.
• Analyze non-null-homologous deformations of contact Anosov flows, showing that semisimplicity persists and a splitting resonance structure appears.
• Incorporate flat unitary twists into both resonant spaces and the Ruelle zeta function via pullback and duality techniques on the unit cotangent bundle.

Experimental results

Research questions

• RQ1 What is the dimension of the resonant space $\text{Res}_k(0)$ for volume-preserving Anosov flows on 3-manifolds, and how does it depend on topological and dynamical invariants?
• RQ2 Under what conditions does the dimension of $\text{Res}_1(0)$ exceed $b_1(M)$, and what role does helicity play?
• RQ3 Does 1-semisimplicity hold for general volume-preserving Anosov flows, and can it fail for time-changed hyperbolic geodesic flows?
• RQ4 How do non-null-homologous deformations of contact Anosov flows affect the resonance structure and semisimplicity?
• RQ5 What is the effect of a flat unitary twist on the resonant spaces and the order of vanishing of the Ruelle zeta function?

Key findings

• For any volume-preserving Anosov flow on a closed 3-manifold, $\dim\text{Res}_0(0) = \dim\text{Res}_2(0) = 1$.
• If $[\iota_X\Omega] \neq 0$, then $\dim\text{Res}_1(0) = b_1(M) - 1$.
• If $[\iota_X\Omega] = 0$, then $\dim\text{Res}_1(0) = b_1(M)$ when $\mathcal{H}(X) \neq 0$, and $b_1(M) + 1$ when $\mathcal{H}(X) = 0$.
• The dimensions of $\text{Res}_k(0)$ coincide with Pollicott-Ruelle resonance multiplicities $m_k(0)$ if and only if the flow is $k$-semisimple.
• There exists a time-changed hyperbolic geodesic flow for which 1-semisimplicity fails, due to the Hölder continuity of the stable and unstable bundles.
• For non-null-homologous deformations of contact Anosov flows, a splitting resonance structure appears on 1-forms, and 1-semisimplicity is preserved.