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Meromorphic extension of $$L$$-functions of Anosov flows and profinite graphs. (English) Zbl 0668.58041
Let $$X$$ be a compact Riemannian manifold and let $$\phi_ t: X\to X$$ be an Anosov flow the non-wandering set of which coincides with $$X$$. The notion of an $$L$$-function $$L_{\phi_ t}(s;\rho)$$ of $$\phi_ t$$ associated with a unitary representation $$\rho: \pi_ 1(X)\to U(N)$$ was defined and studied by the author and T. Sunada [J. Funct. Anal. 71, 1–46 (1987; Zbl 0658.58034)]. It is shown in the paper under review that $$L_{\phi_ t}(s;\rho)$$ extends non-zero meromorphically to some domain $$\text{Re}(s)>h-\delta$$, $$\delta >0$$, where $$h=h(\phi_ t)$$ denotes the topological entropy of the flow $$\phi_ t$$. The proof is an extension of the proof given by M. Pollicott [Invent. Math. 85, 147–164 (1986; Zbl 0604.58042)] for the case when $$\rho$$ is the trivial representation. Similar result is established for the $$L$$-function of a profinite graph associated with a unitary representation of the fundamental group of the graph.
Reviewer: L.N.Stoyanov

MSC:
 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.