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SPECTRAL ANALYSIS OF MORSE–SMALE FLOWS I: CONSTRUCTION OF THE ANISOTROPIC SPACES

Part of: Partial differential equations on manifolds; differential operators Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  13 November 2018

Nguyen Viet Dang  and
Gabriel Rivière
Nguyen Viet Dang
Affiliation:
Institut Camille Jordan (U.M.R. CNRS 5208), Université Claude Bernard Lyon 1, Bâtiment Braconnier, 43, boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France ([email protected])
Gabriel Rivière
Affiliation:
Laboratoire Paul Painlevé (U.M.R. CNRS 8524), U.F.R. de Mathématiques, Université Lille 1, 59655 Villeneuve d’Ascq Cedex, France ([email protected])
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Abstract

We prove the existence of a discrete correlation spectrum for Morse–Smale flows acting on smooth forms on a compact manifold. This is done by constructing spaces of currents with anisotropic Sobolev regularity on which the Lie derivative has a discrete spectrum.

Keywords

transfer operatorsPollicott–Ruelle resonancesMorse–Smale flowsMicrolocal analysis

MSC classification

Primary: 37D15: Morse-Smale systems
Secondary: 58J50: Spectral problems; spectral geometry; scattering theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 5 , September 2020 , pp. 1409 - 1465
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Copyright
© Cambridge University Press 2018

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References

Baladi, V., Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps — A Functional Approach, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/A Series of Modern Surveys in Mathematics, vol. 68 (Springer International Publishing, 2018).Google Scholar
Baladi, V. and Tsujii, M., Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms, Ann. Inst. Fourier (Grenoble) 57 (2007), 127–154.Google Scholar
Baladi, V. and Tsujii, M., Dynamical Determinants and Spectrum for Hyperbolic Diffeomorphisms, Contemporary Mathematics, Volume 469, pp. 29–68 (American Mathematical Society, 2008). Volume in honour of M. Brin’s 60th birthday.Google Scholar
Bowen, R., Symbolic dynamics for hyperbolic flows, Amer. J. Math. 95 (1973), 429–460.Google Scholar
Brin, M. and Stuck, G., Introduction to Dynamical Systems (Cambridge University Press, 2002).Google Scholar
Butterley, O. and Liverani, C., Smooth Anosov flows: correlation spectra and stability, J. Mod. Dyn. 1 (2007), 301–322.Google Scholar
Chen, K. T., Equivalence and decomposition of vector fields about an elementary critical point, Amer. J. Math 85 (1963), 693–722.Google Scholar
Dang, N. V. and Rivière, G., Spectral analysis of Morse–Smale gradient flows, Ann. Sci. ENS, to appear, Preprint, 2016, arXiv:1605.05516.Google Scholar
Dang, N. V. and Rivière, G., Spectral analysis of Morse–Smale flows II: resonances and resonant states, Amer. J. Math., Preprint, 2017, arXiv:1703.08038.Google Scholar
Dang, N. V. and Rivière, G., Topology of Pollicott–Ruelle resonant states, Preprint, 2017, arXiv:1703.08037.Google Scholar
Dyatlov, S. and Guillarmou, C., Pollicott–Ruelle resonances for open systems, Ann. Henri Poincaré 17 (2016), 3089–3146.Google Scholar
Dyatlov, S. and Zworski, M., Dynamical zeta functions for Anosov flows via microlocal analysis, Ann. Sci. ENS 49 (2016), 543–577.Google Scholar
Dyatlov, S. and Zworski, M., Mathematical theory of scattering resonances, avalaible at http://math.mit.edu/∼dyatlov/res/res.pdf, 2016.Google Scholar
Engel, K. J. and Nagel, R., One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Volume 194 (Springer, New York, 2000).Google Scholar
Faure, F., Roy, N. and Sjöstrand, J., Semi-classical approach for Anosov diffeomorphisms and Ruelle resonances, Open Math. J. 1 (2008), 35–81.Google Scholar
Faure, F. and Sjöstrand, J., Upper bound on the density of Ruelle resonances for Anosov flows, Comm. Math. Phys. 308 (2011), 325–364.Google Scholar
Faure, F. and Tsujii, M., The semiclassical zeta function for geodesic flows on negatively curved manifolds, Inv. Math. 208 (2017), 851–998.Google Scholar
Franks, J. M., Homology and Dynamical Systems, CBMS Regional Conference Series in Mathematics, Volume 49 (American Mathematical Society, Providence, RI, 1982). Published for the Conference Board of the Mathematical Sciences, Washington, D.C.Google Scholar
Fried, D., Lefschetz formulas for flows, Contemp. Math. 58(Part III) (1987), 19–69.Google Scholar
Frenkel, E., Losev, A. and Nekrasov, N., Instantons beyond topological theory. I, J. Inst. Math. Jussieu 10(03) (2011), 463–565.Google Scholar
Giulietti, P., Liverani, C. and Pollicott, M., Anosov flows and dynamical zeta functions, Ann. of Math. (2) 178(2) (2013), 687–773.Google Scholar
Gouëzel, S. and Liverani, C., Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties, J. Differential Geom. 79 (2008), 433–477.Google Scholar
Harvey, F. R. and Lawson, H. B., Morse theory and Stokes theorem, in Surveys in Differential Geometry, Volume VII, pp. 259–311 (International Press, 2000).Google Scholar
Harvey, F. R. and Lawson, H. B., Finite volume flows and Morse theory, Ann. of Math. (2) 153 (2001), 1–25.Google Scholar
Helffer, B. and Sjöstrand, J., Résonances en limite semi-classique, Mém. Soc. Math. Fr. (N.S.) 24–25 (1986), 1–228.Google Scholar
Hirsch, M. W., Pugh, C. C. and Shub, M., Invariant Manifolds, Lecture Notes in Mathematics, Volume 583 (Springer, New York, 1977).Google Scholar
Laudenbach, F., On the Thom–Smale complex, in An Extension of a Theorem of Cheeger and Müller (ed. Bismut, J.-M. and Zhang, W.), Astérisque, Volume 205 (Société Math. de France, Paris, 1992).Google Scholar
Liverani, C., On contact Anosov flows, Ann. of Math. (2) 159(3) (2004), 1275–1312.Google Scholar
Lee, J. M., Manifolds and Differential Geometry, Graduate Studies in Mathematics, Volume 107 (American Mathematical Society, Providence Rhode Island, 2009).Google Scholar
Meyer, K. R., Energy functions for Morse–Smale systems, Amer. J. Math. 90 (1968), 1031–1040.Google Scholar
Nelson, E., Topics in Dynamics. I. Flows, Mathematical Notes (University of Tokyo Press, Tokyo, 1969). Princeton University Press, Princeton, NJ, iii+118 pp.Google Scholar
Palis, J., On Morse–Smale dynamical systems, Topology 8 (1968), 385–404.Google Scholar
Palis, J. and de Melo, W., Geometric Theory of Dynamical Systems. An Introduction (Springer, New York, 1982).Google Scholar
Peixoto, M. M., Structural stability on two-dimensional manifolds, Topology 1 (1962), 101–120.Google Scholar
Pollicott, M., On the rate of mixing of Axiom A flows, Invent. Math. 81(3) (1985), 413–426.Google Scholar
de Rham, G., Differentiable Manifolds: Forms, Currents, Harmonic Forms (Springer, 1980).Google Scholar
Ruelle, D., Resonances for axiom a flows, J. Differential Geom. 25(1) (1987), 99–116.Google Scholar
Smale, S., Morse inequalities for a dynamical system, Bull. Amer. Math. Soc. 66 (1960), 43–49.Google Scholar
Smale, S., Differentiable dynamical systems, Bull. Amer. Math. Soc. (N.S.) 73 (1967), 747–817.Google Scholar
Taylor, M., Partial Differential Equations II: Qualitative Studies of Linear Equations, second edition, Applied Mathematical Sciences, vol. 116 (Springer-Verlag, New York).Google Scholar
Teschl, G., Ordinary Differential Equations and Dynamical Systems, Graduate Studies in Mathematics, Volume 140 (American Mathematical Society, Providence Rhode Island, 2012).Google Scholar
Tsujii, M., Quasi-compactness of transfer operators for contact Anosov flows, Nonlinearity 23 (2010), 1495–1545.Google Scholar
Tsujii, M., Contact Anosov flows and the Fourier-Bros-Iagolnitzer transform, Ergodic Theory Dynam. Systems 32(6) (2012), 2083–2118.Google Scholar
Wang, P., Wu, H. and Li, W. G., Normal forms for periodic orbis of real vector fields, Acta Math. Sin. (Engl. Ser.) 24 (2008), 797–808.Google Scholar
Weber, J., The Morse-Witten complex via dynamical systems, Expo. Math. 24 (2006), 127–159.Google Scholar
Zworski, M., Semiclassical Analysis, Graduate Studies in Mathematics, Volume 138 (American Mathematical Society, 2012).Google Scholar