À l’aide des interactions entre les points fixes et les orbites fermées d’un flot de Morse-Smale, nous introduisons une métrique de Milnor sur le déterminant de la cohomologie de la variété fermée sous-jacente à valeurs dans un fibré vectoriel plat. Ceci permet de généraliser la notion de valeur absolue au point zéro de la fonction zêta dynamique de Ruelle, même dans le cas où cette valeur n’est pas bien définie au sens classique. Nous donnons une formule reliant les métriques de Milnor et de Ray-Singer. Un ingrédient essentiel de notre preuve est le théorème de Bismut-Zhang.
We introduce a Milnor metric on the determinant line of the cohomology of the underlying closed manifold with coefficients in a flat vector bundle, by means of interactions between the fixed points and the closed orbits of a Morse-Smale flow. This enables us to generalize the notion of absolute value at the zero point of the Ruelle dynamical zeta function, even in the case where this value is not well-defined in the classical sense. We give a formula relating the Milnor metric and the Ray-Singer metric. An essential ingredient of our proof is Bismut-Zhang’s theorem.
CC-BY 4.0
@article{JEP_2021__8__585_0,
author = {Shu Shen and Jianqing Yu},
title = {Morse-Smale flow, {Milnor} metric, and dynamical zeta function},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {585--607},
publisher = {\'Ecole polytechnique},
volume = {8},
year = {2021},
doi = {10.5802/jep.154},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.154/}
}
TY - JOUR AU - Shu Shen AU - Jianqing Yu TI - Morse-Smale flow, Milnor metric, and dynamical zeta function JO - Journal de l’École polytechnique — Mathématiques PY - 2021 SP - 585 EP - 607 VL - 8 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.154/ DO - 10.5802/jep.154 LA - en ID - JEP_2021__8__585_0 ER -
%0 Journal Article %A Shu Shen %A Jianqing Yu %T Morse-Smale flow, Milnor metric, and dynamical zeta function %J Journal de l’École polytechnique — Mathématiques %D 2021 %P 585-607 %V 8 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.154/ %R 10.5802/jep.154 %G en %F JEP_2021__8__585_0
Shu Shen; Jianqing Yu. Morse-Smale flow, Milnor metric, and dynamical zeta function. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 585-607. doi : 10.5802/jep.154. https://jep.centre-mersenne.org/articles/10.5802/jep.154/
[BG01] - Families torsion and Morse functions, Astérisque, vol. 275, Société Mathématique de France, Paris, 2001 | Numdam | Zbl
[BGS88] - “Analytic torsion and holomorphic determinant bundles. I. Bott-Chern forms and analytic torsion”, Comm. Math. Phys. 115 (1988) no. 1, p. 49-78 | DOI | MR | Zbl
[Bis11] - Hypoelliptic Laplacian and orbital integrals, Annals of Math. Studies, vol. 177, Princeton University Press, Princeton, NJ, 2011 | MR | Zbl
[BM13] - “On the gluing formula for the analytic torsion”, Math. Z. 273 (2013) no. 3-4, p. 1085-1117 | DOI | MR | Zbl
[BZ92] - An extension of a theorem by Cheeger and Müller, Astérisque, vol. 205, Société Mathématique de France, Paris, 1992, With an appendix by François Laudenbach | Numdam
[BZ94] - “Milnor and Ray-Singer metrics on the equivariant determinant of a flat vector bundle”, Geom. Funct. Anal. 4 (1994) no. 2, p. 136-212 | DOI | MR | Zbl
[Che45] - “On the curvatura integra in a Riemannian manifold”, Ann. of Math. (2) 46 (1945), p. 674-684 | DOI | MR | Zbl
[Che79] - “Analytic torsion and the heat equation”, Ann. of Math. (2) 109 (1979) no. 2, p. 259-322 | DOI | MR | Zbl
[dR50] - “Complexes à automorphismes et homéomorphie différentiable”, Ann. Inst. Fourier (Grenoble) 2 (1950), p. 51-67 | Numdam | MR | Zbl
[DR19] - “Spectral analysis of Morse-Smale gradient flows”, Ann. Sci. École Norm. Sup. (4) 52 (2019) no. 6, p. 1403-1458 | DOI | MR | Zbl
[DR20a] - “Spectral analysis of Morse-Smale flows I: construction of the anisotropic spaces”, J. Inst. Math. Jussieu 19 (2020) no. 5, p. 1409-1465 | DOI | MR | Zbl
[DR20b] - “Spectral analysis of Morse-Smale flows, II: Resonances and resonant states”, Amer. J. Math. 142 (2020) no. 2, p. 547-593 | DOI | MR | Zbl
[DR20c] - “Topology of Pollicott-Ruelle resonant states”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. XXI (2020), p. 827-871, special issue | DOI
[DR21] - “Pollicott-Ruelle spectrum and Witten Laplacians”, J. Eur. Math. Soc. (JEMS) (2021), online first | DOI
[Fra35] - “Über die Torsion einer Überdeckung”, J. reine angew. Math. 173 (1935), p. 245-254 | DOI | Zbl
[Fra79] - “Morse-Smale flows and homotopy theory”, Topology 18 (1979) no. 3, p. 199-215 | DOI | MR | Zbl
[Fra82] - Homology and dynamical systems, CBMS Regional Conference Series in Math., vol. 49, American Mathematical Society, Providence, RI, 1982 | MR | Zbl
[Fri86] - “Analytic torsion and closed geodesics on hyperbolic manifolds”, Invent. Math. 84 (1986) no. 3, p. 523-540 | DOI | MR | Zbl
[Fri87] - “Lefschetz formulas for flows”, in The Lefschetz centennial conference, Part III (Mexico City, 1984), Contemp. Math., vol. 58, American Mathematical Society, Providence, RI, 1987, p. 19-69 | DOI | MR | Zbl
[Hör90] - The analysis of linear partial differential operators. I, Grundlehren Math. Wiss., vol. 256, Springer-Verlag, Berlin, 1990 | DOI | MR
[KM76] - “The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div””, Math. Scand. 39 (1976) no. 1, p. 19-55 | DOI | MR | Zbl
[KT74] - “Characteristic invariants of foliated bundles”, Manuscripta Math. 11 (1974), p. 51-89 | DOI | MR | Zbl
[LR91] - “Analytic torsion for group actions”, J. Differential Geom. 34 (1991) no. 2, p. 431-481 | DOI | MR | Zbl
[Lüc93] - “Analytic and topological torsion for manifolds with boundary and symmetry”, J. Differential Geom. 37 (1993) no. 2, p. 263-322 | DOI | MR | Zbl
[Mil66] - “Whitehead torsion”, Bull. Amer. Math. Soc. 72 (1966), p. 358-426 | DOI | MR | Zbl
[Mil68] - “Infinite cyclic coverings”, in Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, Mich., 1967), Prindle, Weber & Schmidt, Boston, Mass., 1968, p. 115-133 | Zbl
[MQ86] - “Superconnections, Thom classes, and equivariant differential forms”, Topology 25 (1986) no. 1, p. 85-110 | DOI | MR | Zbl
[MS91] - “-torsion and zeta functions for locally symmetric manifolds”, Invent. Math. 105 (1991) no. 1, p. 185-216 | DOI | MR | Zbl
[Mül78] - “Analytic torsion and -torsion of Riemannian manifolds”, Adv. in Math. 28 (1978) no. 3, p. 233-305 | DOI | MR | Zbl
[Mül93] - “Analytic torsion and -torsion for unimodular representations”, J. Amer. Math. Soc. 6 (1993) no. 3, p. 721-753 | DOI | MR | Zbl
[Pal68] - “On Morse-Smale dynamical systems”, Topology 8 (1968), p. 385-404 | DOI | MR
[PdM82] - Geometric theory of dynamical systems, Springer-Verlag, New York-Berlin, 1982
[PS70] - “Structural stability theorems”, in Global Analysis (Berkeley, Calif., 1968), Proc. Sympos. Pure Math., vol. XIV, American Mathematical Society, Providence, RI, 1970, p. 223-231 | Zbl
[Qui85] - “Superconnections and the Chern character”, Topology 24 (1985) no. 1, p. 89-95 | DOI | MR | Zbl
[Rei35] - “Homotopieringe und Linsenräume”, Abh. Math. Sem. Univ. Hamburg 11 (1935) no. 1, p. 102-109 | DOI | MR | Zbl
[RS71] - “-torsion and the Laplacian on Riemannian manifolds”, Adv. in Math. 7 (1971), p. 145-210 | DOI | MR | Zbl
[See67] - “Complex powers of an elliptic operator”, in Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), American Mathematical Society, Providence, RI, 1967, p. 288-307 | DOI | Zbl
[She18] - “Analytic torsion, dynamical zeta functions, and the Fried conjecture”, Anal. PDE 11 (2018) no. 1, p. 1-74 | DOI | MR | Zbl
[SM96] - “Reidemeister torsion and Morse-Smale flows”, Ergodic Theory Dynam. Systems 16 (1996) no. 2, p. 405-414 | DOI | MR | Zbl
[Sma61] - “On gradient dynamical systems”, Ann. of Math. (2) 74 (1961), p. 199-206 | DOI | MR | Zbl
[SY17] - “Flat vector bundles and analytic torsion on orbifolds”, 2017, to appear in Comm. Anal. Geom. | arXiv
Cité par Sources :