Dynamical torsion for contact Anosov flows

Abstract

We introduce a new object, called dynamical torsion, which extends the potentially ill-defined value at 0 of the Ruelle zeta function ζ of a contact Anosov flow twisted by an acyclic representation of the fundamental group. The dynamical torsion depends analytically on the representation and is invariant under deformations among contact Anosov flows. Moreover, we show that the ratio between this torsion and the refined combinatorial torsion of Turaev, for an appropriate choice of Euler structure, is locally constant on the space of acyclic representations. In particular, for contact Anosov flows path connected to a geodesic flow of a hyperbolic manifold among contact Anosov flows, we relate the leading term of the Laurent expansion of ζ at the origin, the Reidemeister torsion and the torsions of the finite dimensional complexes of the generalized resonant states of both flows for the resonance 0. This extends previous work of [DGRS18] on the Fried conjecture near geodesic flows of hyperbolic 3-manifolds, to hyperbolic manifolds of any odd dimensions.

A Morse complex for Axiom A flows

“…Thanks to our choice of a total order relation in the sence of §2.5.2, we have necessarily k ≥ j. It proves the convergence (14). Up to replacing the flow ϕ t by ϕ −t , we also get the convergence (15).…”

“…It allowed them to prove Fried conjecture on the Reidemeister torsion for nearly hyperbolic 3-manifolds. This was further pursued by Chaubet and Dang [14] who used the cohomological complex of Theorem 1 to define a dynamical torsion for contact Anosov flows in any dimension. • Morse-Smale flows.…”

Ruelle zeta function from field theory

“…From the point of view of algebraic topology, this result suggests that, under certain assumptions, ι X can be made into a chain contraction for the de Rham complex, namely one can construct η X = (L X ) −1 ι X with the appropriate conditions of nondegeneracy of L X . This intepretation appears to be related with the notion of a dynamical torsion introduced in [CD19]. There appears to be a sweet spot at the intersection of Anosov and Reeb vector fields where the independence of the "torsion" of the de Rham complex on the choice of a chain contraction, and independence of the partition function of BF theory on a choice of gauge fixing appear to be aspects of the same statement, expressed by Fried's conjecture.…”

“…A more precise version for locally symmetric manifolds has been proved in [She17] following [MS91]. In the variable curvature case a perturbative result has been obtained in [DGRS18] and extended in [CD19]. A surprising result in the case of surfaces with variable negative curvature, but without reference to an acyclic representation, showed the zeta function at zero is determined by the topology of the surface [DZ17].…”

The twisted Ruelle zeta function on compact hyperbolic orbisurfaces and Reidemeister-Turaev torsion

“…Moreover, Müller ([19]), Shen ([23]) and the third author ( [25]) proved Fried's conjecture for more general non-unitary representations, with the additional assumption that they are close to a unitary and acyclic representation in the representation variety. In a completely general setting, Chaubet-Dang established in [4] a variational formula relating the value at zero of the Ruelle zeta function to the Reidemeister-Turaev torsion. As a consequence of their work together with our Theorem A we have the following corollary: Corollary B.…”