Zeta functions and the Fried conjecture for smooth pseudo-Anosov flows

2024/09/25 by Malo Jézéquel, Jézéquel, Malo, Jonathan Zung +1
Economics, Econometrics and Finance · Mathematics · #37C30 #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Stochastic processes and financial applications #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2409.17014

openalex publication_date 2024/09/25 · openalex created_date 2024/09/26 · openalex updated_date 2026/07/30

Abstract

To a transitive pseudo-Anosov flow φ on a 3-manifold M and a representation ρ of π₁(M), we associate a zeta function ζ_φ,ρ(s) defined for \Re s ≫ 1, generalizing the Anosov case. For a class of ``smooth pseudo-Anosov flows'', we prove that ζ_φ,ρ(s) has a meromorphic continuation to ℂ. We also prove a version of the Fried conjecture for smooth pseudo-Anosov flows which, under some conditions on ρ, relates ζ_φ,ρ(0) to the Reidemeister torsion of M. Finally we prove a topological analogue of the Dirichlet class number formula. In order to deal with singularities, we use C^∞ versions of the approaches of Rugh and Sanchez--Morgado, based on Markov partitions.

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