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

Abstract

With the help of interactions between the fixed points and the closed orbits of a Morse-Smale flow, we introduce a Milnor metric on the determinant line of the cohomology of the underlying closed manifold with coefficients in a flat vector bundle. This allows us to generalise the notion of the absolute value at 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. CONTENTSIntroduction 1. Preliminary 1.1. The determinant line 1.2. The cohomology of S 1 1.3. A fusion principle 2. Milnor metric 2.1. Morse-Smale flow 2.2. Ruelle dynamical zeta function 2.3. Frank's Morse function 2.4. Smale filtration and Milnor metric 2.5. A comparison formula for Milnor metrics 3. An extension of Bismut-Zhang's Theorem to Morse-Smale flow 3.1. Berezin integral 3.2. Mathai-Quillen formalism 3.3. Ray-Singer metric 3.4. Proof of the main result References

Analytic torsion, dynamical zeta functions, and the Fried conjecture

“…We refer the reader to [ShY18,DaGRSh20] for the Fried conjecture for the Morse-Smale flow and the Anosov flow, to [Sh19] for a survey on the Fried conjecture.…”

Complex valued analytic torsion and dynamical zeta function on locally symmetric spaces

“…We refer the reader to [Sh19] for a survey on the Fried conjecture on different flows, like Morse-Smale flow [ShY18] and Anosov flow [DGRSh20]. Now, we will describe our results in more detail, and explain the techniques used in their proofs.…”