Morse-Smale flow, Milnor metric, and dynamical zeta function

Shen, Shu; Yu, Jianqing

doi:10.5802/jep.154

Morse-Smale flow, Milnor metric, and dynamical zeta function
[Flot de Morse-Smale, métrique de Milnor et fonction zêta dynamique]

Shen, Shu ¹ ; Yu, Jianqing ²

¹ Institut de Mathématiques de Jussieu-Paris Rive Gauche, Sorbonne Université 4 place Jussieu, 75252 Paris Cedex 05, France
² School of Mathematical Sciences, University of Science and Technology of China 96 Jinzhai Road, Hefei, Anhui 230026, P. R. China

Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 585-607.

Résumé
Abstract

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.

Reçu le : 2020-06-12
Accepté le : 2021-02-08
Publié le : 2021-03-05

DOI : 10.5802/jep.154

Classification : 37D15, 37C30, 58J52, 57Q10
Keywords: Index theory and related fixed point theorems, analytic torsion, Selberg trace formula, dynamical zeta functions
Mot clés : Flot de Morse-Smale, fonction zêta dynamique, torsion analytique, fibré déterminant

Affiliations des auteurs :

Shen, Shu ¹ ; Yu, Jianqing ²

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Shen, Shu; Yu, Jianqing. 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. http://archive.numdam.org/articles/10.5802/jep.154/

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