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Resonant forms at zero for dissipative Anosov flows

Mihajlo Cekić and Gabriel P Paternain

Geometry & Topology 29 (2025) 3635–3716
Bibliography
1 D V Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov. 90 (1967) 209 MR224110
2 V I Arnold, B A Khesin, Topological methods in hydrodynamics, 125, Springer (1998) MR1612569
3 V Baladi, Anisotropic Sobolev spaces and dynamical transfer operators : C foliations, from: "Algebraic and topological dynamics" (editors S Kolyada, Y Manin, T Ward), Contemp. Math. 385, Amer. Math. Soc. (2005) 123 MR2180233
4 V Baladi, M Tsujii, Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms, Ann. Inst. Fourier (Grenoble) 57 (2007) 127 MR2313087
5 M Blank, G Keller, C Liverani, Ruelle–Perron–Frobenius spectrum for Anosov maps, Nonlinearity 15 (2002) 1905 MR1938476
6 Y G Bonthonneau, Perturbation of Ruelle resonances and Faure–Sjöstrand anisotropic space, Rev. Un. Mat. Argentina 61 (2020) 63 MR4126207
7 M Cekić, T Lefeuvre, Generic injectivity of the X-ray transform, J. Differential Geom. 129 (2025) 261 MR4859113
8 M Cekić, G P Paternain, Resonant spaces for volume-preserving Anosov flows, Pure Appl. Anal. 2 (2020) 795 MR4229226
9 M Cekić, G P Paternain, Quasi-Fuchsian flows and the coupled vortex equations, preprint (2025) arXiv:2501.10591
10 S Coles, R Sharp, Helicity, linking and the distribution of null-homologous periodic orbits for Anosov flows, Nonlinearity 36 (2023) 21 MR4512749
11 G Contreras, Average linking numbers of closed orbits of hyperbolic flows, J. London Math. Soc. 51 (1995) 614 MR1332896
12 N V Dang, G Rivière, Poincaré series and linking of Legendrian knots, Duke Math. J. 173 (2024) 1 MR4728176
13 S Dyatlov, M Zworski, Dynamical zeta functions for Anosov flows via microlocal analysis, Ann. Sci. Éc. Norm. Supér. 49 (2016) 543 MR3503826
14 S Dyatlov, M Zworski, Ruelle zeta function at zero for surfaces, Invent. Math. 210 (2017) 211 MR3698342
15 S Dyatlov, M Zworski, Mathematical theory of scattering resonances, 200, Amer. Math. Soc. (2019) MR3969938
16 S Dyatlov, F Faure, C Guillarmou, Power spectrum of the geodesic flow on hyperbolic manifolds, Anal. PDE 8 (2015) 923 MR3366007
17 F Faure, J Sjöstrand, Upper bound on the density of Ruelle resonances for Anosov flows, Comm. Math. Phys. 308 (2011) 325 MR2851145
18 F Faure, M Tsujii, Micro-local analysis of contact Anosov flows and band structure of the Ruelle spectrum, Commun. Am. Math. Soc. 4 (2024) 641 MR4840241
19 J Franks, B Williams, Anomalous Anosov flows, from: "Global theory of dynamical systems" (editors Z Nitecki, C Robinson), Lecture Notes in Math. 819, Springer (1980) 158 MR591182
20 É Ghys, Déformations de flots d’Anosov et de groupes fuchsiens, Ann. Inst. Fourier (Grenoble) 42 (1992) 209 MR1162561
21 É Ghys, Rigidité différentiable des groupes fuchsiens, Inst. Hautes Études Sci. Publ. Math. 78 (1993) 163 MR1259430
22 P Giulietti, C Liverani, M Pollicott, Anosov flows and dynamical zeta functions, Ann. of Math. 178 (2013) 687 MR3071508
23 C Guillarmou, F Faure, Horocyclic invariance of Ruelle resonant states for contact Anosov flows in dimension 3, Math. Res. Lett. 25 (2018) 1405 MR3917733
24 C Guillarmou, J Hilgert, T Weich, Classical and quantum resonances for hyperbolic surfaces, Math. Ann. 370 (2018) 1231 MR3770166
25 R S Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. 7 (1982) 65 MR656198
26 B Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations, Ergodic Theory Dynam. Systems 14 (1994) 645 MR1304137
27 L Hörmander, Pseudo-differential operators, Comm. Pure Appl. Math. 18 (1965) 501 MR180740
28 L Hörmander, The analysis of linear partial differential operators, I : Distribution theory and Fourier analysis, Springer (2003) MR1996773
29 L Hörmander, The analysis of linear partial differential operators, III : Pseudo-differential operators, Springer (2007) MR2304165
30 S Hurder, A Katok, Differentiability, rigidity and Godbillon–Vey classes for Anosov flows, Inst. Hautes Études Sci. Publ. Math. 72 (1990) 5 MR1087392
31 D Kotschick, T Vogel, Linking numbers of measured foliations, Ergodic Theory Dynam. Systems 23 (2003) 541 MR1972238
32 B Küster, T Weich, Pollicott–Ruelle resonant states and Betti numbers, Comm. Math. Phys. 378 (2020) 917 MR4134938
33 S P Lalley, Distribution of periodic orbits of symbolic and Axiom A flows, Adv. in Appl. Math. 8 (1987) 154 MR886923
34 S Lang, Fundamentals of differential geometry, 191, Springer (1999) MR1666820
35 J M Lee, Introduction to Riemannian manifolds, 176, Springer (2018) MR3887684
36 C Liverani, On contact Anosov flows, Ann. of Math. 159 (2004) 1275 MR2113022
37 R de la Llave, J M Marco, R Moriyón, Canonical perturbation theory of Anosov systems and regularity results for the Livšic cohomology equation, Ann. of Math. 123 (1986) 537 MR840722
38 T Marty, Skewed Anosov flows are orbit equivalent to Reeb–Anosov flows in dimension 3, preprint (2023) arXiv:2301.00842
39 R B Melrose, Introduction to microlocal analysis, book draft (2003)
40 W Merry, G P Paternain, Inverse problems in geometry and dynamics, lecture notes (2011)
41 U Neri, The integrable kernels of certain pseudo-differential operators, Math. Ann. 186 (1970) 155 MR261143
42 G P Paternain, Regularity of weak foliations for thermostats, Nonlinearity 20 (2007) 87 MR2285106
43 J F Plante, Anosov flows, transversely affine foliations, and a conjecture of Verjovsky, J. London Math. Soc. 23 (1981) 359 MR609116
44 P Przytycki, M P Wojtkowski, Gaussian thermostats as geodesic flows of nonsymmetric linear connections, Comm. Math. Phys. 277 (2008) 759 MR2365452
45 D Ruelle, Positivity of entropy production in nonequilibrium statistical mechanics, J. Statist. Phys. 85 (1996) 1 MR1413234
46 D Ruelle, Differentiation of SRB states for hyperbolic flows, Ergodic Theory Dynam. Systems 28 (2008) 613 MR2408395
47 S Schwartzman, Asymptotic cycles, Ann. of Math. 66 (1957) 270 MR88720
48 R Sharp, Prime orbit theorems with multi-dimensional constraints for Axiom A flows, Monatsh. Math. 114 (1992) 261 MR1203975
49 R Sharp, Closed orbits in homology classes for Anosov flows, Ergodic Theory Dynam. Systems 13 (1993) 387 MR1235480
50 I M Singer, J A Thorpe, Lecture notes on elementary topology and geometry, Springer (1976) MR413152
51 M E Taylor, Partial differential equations, III : Nonlinear equations, 117, Springer (1997) MR1477408
52 M Tsujii, On cohomological theory of dynamical zeta functions, preprint (2018) arXiv:1805.11992
53 T Vogel, On the asymptotic linking number, Proc. Amer. Math. Soc. 131 (2003) 2289 MR1963779
54 T Weich, On the support of Pollicott–Ruelle resonanant states for Anosov flows, Ann. Henri Poincaré 18 (2017) 37 MR3592089
55 M P Wojtkowski, W-flows on Weyl manifolds and Gaussian thermostats, J. Math. Pures Appl. 79 (2000) 953 MR1801870
56 L S Young, What are SRB measures, and which dynamical systems have them ?, J. Statist. Phys. 108 (2002) 733 MR1933431
57 M Zworski, Semiclassical analysis, 138, Amer. Math. Soc. (2012) MR2952218