Resonant forms at zero for dissipative Anosov flows
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866908592254222336 |
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| author | Cekić, Mihajlo Paternain, Gabriel P. |
| author_facet | Cekić, Mihajlo Paternain, Gabriel P. |
| contents | 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 $-χ(M)-1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_06255 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Resonant forms at zero for dissipative Anosov flows Cekić, Mihajlo Paternain, Gabriel P. Dynamical Systems Analysis of PDEs Differential Geometry Geometric Topology Spectral Theory 37C30, 37D40, 58C40, 53C65, 35P05 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 $-χ(M)-1$. |
| title | Resonant forms at zero for dissipative Anosov flows |
| topic | Dynamical Systems Analysis of PDEs Differential Geometry Geometric Topology Spectral Theory 37C30, 37D40, 58C40, 53C65, 35P05 |
| url | https://arxiv.org/abs/2211.06255 |