The Fried Conjecture for Morse-Smale Flows: A Survey on Ray-Singer and Milnor Metrics
Abstract
This paper reviews the generalised Fried conjecture for Morse--Smale flows on compact Riemannian manifolds. We first establish the proper framework by constructing the twisted de Rham complex and deriving the Hodge decomposition, which underpins the definition of the Ray--Singer torsion. On the dynamical side, we characterise Morse--Smale vector fields, introducing the Ruelle Zeta function to encode the spectral data of closed orbits and {reviewing the Thom--Smale complex for purely gradient flows. In general, the local contributions of the critical elements of the flow are globally assembled via a Smale filtration to construct the Milnor metric} on the determinant line of the twisted cohomology. Finally, under the assumption of a unitary representation, we present the theorem proving the conjecture: the Ray--Singer metric coincides with the Milnor metric. Furthermore, we discuss the structure of the Milnor metric, which factors into the Ruelle Zeta function evaluated at zero and a specific contribution from the fixed points. We clarify that this object reduces to either the classical algebraic Thom--Smale torsion or the Ruelle Zeta function solely in the limiting cases of purely gradient or fixed-point-free flows.
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BibTeX
@misc{molinari2026fried,
title = {{The Fried Conjecture for Morse-Smale Flows: A Survey on Ray-Singer and Milnor Metrics}},
author = {Giovanni Molinari},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.16719},
eprint = {2607.16719},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.16719},
}
RIS
TY - UNPB
TI - The Fried Conjecture for Morse-Smale Flows: A Survey on Ray-Singer and Milnor Metrics
AU - Giovanni Molinari
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2607.16719
UR - https://arxiv.org/abs/2607.16719
AB - This paper reviews the generalised Fried conjecture for Morse--Smale flows on compact Riemannian manifolds. We first establish the proper framework by constructing the twisted de Rham complex and deriving the Hodge decomposition, which underpins the definition of the Ray--Singer torsion. On the dynamical side, we characterise Morse--Smale vector fields, introducing the Ruelle Zeta function to encode the spectral data of closed orbits and {reviewing the Thom--Smale complex for purely gradient flows. In general, the local contributions of the critical elements of the flow are globally assembled via a Smale filtration to construct the Milnor metric} on the determinant line of the twisted cohomology. Finally, under the assumption of a unitary representation, we present the theorem proving the conjecture: the Ray--Singer metric coincides with the Milnor metric. Furthermore, we discuss the structure of the Milnor metric, which factors into the Ruelle Zeta function evaluated at zero and a specific contribution from the fixed points. We clarify that this object reduces to either the classical algebraic Thom--Smale torsion or the Ruelle Zeta function solely in the limiting cases of purely gradient or fixed-point-free flows.
ER -